harmonisation standard UQO WP (2026-10-17)

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>

Simon-Pierre Boucher committed 2 h ago (Oct 17, 2026)

44 changed files +5,560 −0

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	1	+# Raw & processed data — never committed (see data/raw/README.md for provenance)
	2	+data/raw/*
	3	+!data/raw/README.md
	4	+data/processed/*
	5	+!data/processed/.gitkeep
	6	+
	7	+# Python
	8	+__pycache__/
	9	+*.pyc
	10	+.pytest_cache/
	11	+.venv/
	12	+*.egg-info/
	13	+
	14	+# LaTeX build artifacts (paper/main.pdf IS committed)
	15	+paper/*.aux
	16	+paper/*.log
	17	+paper/*.out
	18	+paper/*.toc
	19	+paper/*.lof
	20	+paper/*.lot
	21	+paper/*.fls
	22	+paper/*.fdb_latexmk
	23	+paper/*.bbl
	24	+paper/*.blg
	25	+paper/*.spl
	26	+paper/*.synctex.gz
	27	+paper/*.run.xml
	28	+paper/*.bcf
	29	+
	30	+# Logs
	31	+logs/
	32	+*.log
	33	+
	34	+# macOS
	35	+.DS_Store
	36	+
	37	+# Project-specific: Centris listings snapshot (JSON, ~18k–42k listings)
	38	+*.json

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	1	+<!-- Author: Simon-Pierre Boucher — simon-pierre.boucher@uqo.ca -->
	2	+# 🏡 Heterogeneous Amenity Capitalization in the Quebec Residential Market
	3	+
	4	+UQO Working Paper No. 1 — A Spatial Quantile Regression Approach
	5	+
	6	+[![Paper](https://img.shields.io/badge/paper-PDF%20(49%20p.)-b31b1b?logo=latex&logoColor=white)](paper/main.pdf)
	7	+[![Python](https://img.shields.io/badge/python-3.11%2B-3776AB?logo=python&logoColor=white)](requirements.txt)
	8	+[![Data](https://img.shields.io/badge/data-42%2C481%20Centris%20listings%20·%20JSON-4051b5)](data/raw/README.md)
	9	+[![Reproducible](https://img.shields.io/badge/reproducible-end--to--end%20pipeline-2e7d32)](scripts/)
	10	+[![Institution](https://img.shields.io/badge/UQO-D%C3%A9pt.%20des%20sciences%20administratives-16365c)](https://uqo.ca)
	11	+[![Author](https://img.shields.io/badge/author-Simon--Pierre%20Boucher-16365c)](mailto:simon-pierre.boucher@uqo.ca)
	12	+
	13	+> TL;DR — Do neighbourhood amenities capitalize into house prices the same way at the bottom and at the top of the
	14	+> price distribution? Using 42,481 Centris listings in four segments (houses, condos, plexes, land), we estimate quantile
	15	+> regressions at τ ∈ {.10, .25, .50, .75, .90} augmented with an instrumented spatial lag (two-stage spatial quantile
	16	+> regression, 2SQR). Coefficient homogeneity across quantiles is rejected (omnibus χ²(7) = 35.7, p < 0.001 for houses):
	17	+> park proximity is worth +8.7% at the 10th percentile and nothing at the 90th; hospital proximity is a −8.2% discount at
	18	+> the bottom and vanishes at the top. The spatial lag ρ is positive and significant in every segment (Moran's I on OLS
	19	+> residuals 0.44–0.55). First multi-segment spatial quantile analysis of amenity capitalization for Quebec.
	20	+
	21	+---
	22	+
	23	+## 📌 Status
	24	+
	25	+\| Item \| Value \|
	26	+\|---\|---\|
	27	+\| Version \| 1.0 (May 2026) \|
	28	+\| SSRN \| not yet submitted \|
	29	+\| Journal \| — \|
	30	+\| Code & data \| reproducible from `scripts/`; the Centris snapshot is not redistributed (see `data/raw/README.md`) \|
	31	+
	32	+## 🎯 What this paper does
	33	+
	34	+Standard hedonic models estimated by OLS impose a single implicit price for every attribute, whatever the price segment.
	35	+This paper asks whether the capitalization of neighbourhood amenities (hospitals, parks, transit, schools, shopping) varies
	36	+along the conditional price distribution of Quebec residential listings, and whether that heterogeneity survives once
	37	+spatial dependence is taken into account.
	38	+
	39	+```
	40	+Q_τ(ln P_i \| X_i, W ln P) = α(τ) + ρ(τ) · [W ln P]_i + X_i'β(τ) τ ∈ {.10, .25, .50, .75, .90}
	41	+```
	42	+
	43	+The spatial lag `W ln P` is instrumented by spatially lagged covariates (Kim & Muller, 2004), standard errors are
	44	+bootstrapped (200 replications), inter-quantile Wald tests probe β(τ_low) = β(τ_high), and Moran's I on residuals
	45	+diagnoses remaining spatial autocorrelation. The analysis is run separately for single-family houses, condominiums,
	46	+multiplexes and vacant land.
	47	+
	48	+## 🏆 Headline results
	49	+
	50	+\| Object \| Estimate \|
	51	+\|---\|---\|
	52	+\| Omnibus Wald test of coefficient homogeneity (houses) \| χ²(7) = 35.74, p < 0.001 \|
	53	+\| Hospital proximity, τ = .10 → .90 \| −0.082 (sig.) → positive, insignificant \|
	54	+\| Park proximity, τ = .10 → .90 \| +0.087 (p < 0.001) → −0.044 (n.s.) \|
	55	+\| Transit proximity, τ = .10 → .90 \| +0.074 → −0.008 \|
	56	+\| Spatial lag ρ (τ = .50) \| 0.29 (houses), 0.20 (condos); positive and significant in all segments \|
	57	+\| Moran's I on OLS residuals \| 0.44 – 0.55 across segments \|
	58	+\| First-stage F-statistics \| > 100 in every segment \|
	59	+
	60	+Interpretation: listing prices and cross-sectional variation only, so coefficients are quantile-specific capitalization
	61	+gradients, not causal effects.
	62	+
	63	+## 🗂 Repository layout
	64	+
	65	+```
	66	+amenity-capitalization-spatial-quantile/
	67	+├── data/
	68	+│ ├── raw/ # Centris JSON snapshot (see data/raw/README.md) — not committed
	69	+│ └── processed/
	70	+├── scripts/ # the pipeline, in order
	71	+│ ├── 01_hedonic_ols_v1.py # baseline log-linear OLS, single-family subset
	72	+│ ├── 02_hedonic_ols_v2.py # quadratic terms, surface selection correction, spatial interactions
	73	+│ ├── 03_hedonic_ols_final.py # final OLS specification (centred variables, HC3)
	74	+│ ├── 04_quantile_spatial_hedonic.py # QR + instrumented spatial lag, Moran, Wald (houses)
	75	+│ └── 05_sqr_multi_segment.py # full multi-segment 2SQR with bootstrap, figures, tables (the paper)
	76	+├── results/
	77	+│ └── sqr_multi_segment_output.txt # complete console output of step 05
	78	+├── figures/ # fig2_wtp_by_quantile.png (others regenerated by step 05)
	79	+└── paper/ # main.tex + sections/ + tables/ → main.pdf
	80	+```
	81	+
	82	+## 🚀 Quick start
	83	+
	84	+```bash
	85	+python3 -m pip install -r requirements.txt
	86	+# put the Centris JSON snapshot in data/raw/, then:
	87	+python3 scripts/03_hedonic_ols_final.py # OLS benchmark
	88	+python3 scripts/05_sqr_multi_segment.py data/raw/house_listings_1767845793472.json # 2SQR, 4 segments, figures fig1–fig7, LaTeX tables
	89	+cd paper && make # latexmk → main.pdf
	90	+```
	91	+
	92	+Runtime ≈ 15 minutes on an Apple-silicon laptop (bootstrap 200 × 5 quantiles × 4 segments).
	93	+
	94	+## 🗃 Data
	95	+
	96	+42,481 Centris listings (Quebec, May 2026) with asking price, structural characteristics, coordinates and distances to
	97	+amenities; four segments. Listings with missing price, area or coordinates are dropped; prices are winsorized at 1/99%.
	98	+Details and access conditions in `data/raw/README.md`.
	99	+
	100	+## 🔬 Methodology
	101	+
	102	+1. OLS hedonic benchmark (log-linear, HC3) — `scripts/03_hedonic_ols_final.py`.
	103	+2. Quantile regression (Koenker & Bassett, 1978) at five quantiles — `scripts/04_…`, `scripts/05_…`.
	104	+3. Two-stage spatial quantile regression: spatial lag of log price built from a k-nearest-neighbour weight matrix,
	105	+ instrumented by lagged covariates (Kim & Muller, 2004) — `scripts/05_sqr_multi_segment.py`.
	106	+4. Inference: 200-replication bootstrap standard errors; inter-quantile Wald tests; Moran's I on residuals;
	107	+ sensitivity to k (number of neighbours).
	108	+5. Willingness-to-pay by quantile, direct/indirect effects at τ = .50.
	109	+
	110	+## 🖼 Figures & tables inventory
	111	+
	112	+\| Figure \| Content \|
	113	+\|---\|---\|
	114	+\| `fig1_coef_plots_amenities` \| amenity coefficients across quantiles, by segment \|
	115	+\| `fig2_wtp_by_quantile` \| willingness to pay by quantile \|
	116	+\| `fig3_ols_vs_qr_vs_sqr` \| OLS vs QR vs 2SQR coefficients \|
	117	+\| `fig4_pseudo_r2` \| pseudo-R² by quantile \|
	118	+\| `fig5_spatial_lag_rho` \| spatial lag ρ(τ) \|
	119	+\| `fig6_price_distributions` \| price distributions by segment \|
	120	+\| `fig7_wald_heatmap` \| inter-quantile Wald tests \|
	121	+
	122	+Tables (`paper/tables/*.tex`): sample, variables, descriptives (houses / other segments), specification, QR houses,
	123	+SQR ρ, first stage, Moran (OLS / SQR), Wald, WTP, effects, amenity correlations, sensitivity to k.
	124	+
	125	+## ⚠️ Limitations
	126	+
	127	+- Listing (asking) prices, not transaction prices; cross-section only.
	128	+- Amenity distances are straight-line; no quality adjustment of amenities.
	129	+- Spatial weights based on k nearest neighbours; results are checked for k but not for alternative kernels.
	130	+- Quantile-specific coefficients are capitalization gradients, not causal effects.
	131	+
	132	+## 📝 Citation
	133	+
	134	+```bibtex
	135	+@techreport{boucher2026amenity,
	136	+ author = {Boucher, Simon-Pierre},
	137	+ title = {Heterogeneous Amenity Capitalization in the Quebec Residential Market: A Spatial Quantile Regression Approach},
	138	+ institution = {Universit\'e du Qu\'ebec en Outaouais, D\'epartement des sciences administratives},
	139	+ type = {Working Paper},
	140	+ number = {1},
	141	+ year = {2026},
	142	+ month = {May}
	143	+}
	144	+```
	145	+
	146	+## 👤 Author & contact
	147	+
	148	+Simon-Pierre Boucher — Département des sciences administratives,
	149	+Université du Québec en Outaouais (UQO), Gatineau, QC.
	150	+📧 [simon-pierre.boucher@uqo.ca](mailto:simon-pierre.boucher@uqo.ca)
	151	+
	152	+## 🗒 Notes
	153	+
	154	+- Bibliography is an inline `thebibliography` (`paper/sections/references.tex`, 103 entries), not a `.bib` file —
	155	+ kept as is (deviation from `Recherche-UQO/HARMONISATION.md` §3).
	156	+- Figures 1 and 3–7 are not stored in the repository; `paper/main.tex` prints a placeholder box for a missing figure.
	157	+ Run `scripts/05_sqr_multi_segment.py` with the data snapshot and copy the PNGs to `paper/figures/` to restore them.
	158	+- The original flat LaTeX draft (`paper_sqr_hedonic.tex`, May 2026) is archived in
	159	+ `Recherche-UQO/_archives/versions-mai-2026/amenity-capitalization--flat-tex-20260519/`.
	160	+- Cover date corrected from "May 2025" (typo) to May 2026 during the 2026-10-17 harmonisation.

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	1	+# Raw data — Centris listings snapshot
	2	+
	3	+\| Item \| Value \|
	4	+\|---\|---\|
	5	+\| Source \| Centris.ca residential listings (Quebec), scraped snapshot \|
	6	+\| File expected here \| `house_listings_1767845793472.json` (JSON array, one object per listing) \|
	7	+\| Observations \| 42,481 listings across four segments (single-family houses, condominiums, plexes, vacant land); the earlier OLS scripts (01–03) use the 18,920 single-family subset \|
	8	+\| Extraction \| May 2026 \|
	9	+\| Fields used \| asking price, living area, lot size, bedrooms, bathrooms, year built, property type, latitude/longitude, distances to amenities (hospital, park, transit, school, shopping), municipality \|
	10	+\| Access \| Not redistributed (Centris terms of use). Place the JSON file in this folder; `scripts/05_sqr_multi_segment.py` accepts the path as its first argument. \|

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	1	+$pdf_mode = 1;
	2	+$pdflatex = 'pdflatex -interaction=nonstopmode -halt-on-error -synctex=1 %O %S';
	3	+$bibtex_use = 2;
	4	+$clean_ext = 'synctex.gz run.xml bbl bcf fdb_latexmk fls log aux out toc lof lot blg';
	5	+@default_files = ('main.tex');

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	1	+.PHONY: all clean distclean
	2	+
	3	+all:
	4	+ latexmk
	5	+
	6	+clean:
	7	+ latexmk -c
	8	+
	9	+distclean:
	10	+ latexmk -C
	11	+ rm -f .synctex.gz .run.xml .bbl .bcf

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	1	+% ============================================================================
	2	+% UQO Working Paper No. 1
	3	+% Heterogeneous Amenity Capitalization in the Quebec Residential Market:
	4	+% A Spatial Quantile Regression Approach
	5	+% ============================================================================
	6	+\documentclass[12pt,letterpaper]{article}
	7	+
	8	+% --- Encoding & Language ---
	9	+\usepackage[utf8]{inputenc}
	10	+\usepackage[T1]{fontenc}
	11	+\usepackage[english]{babel}
	12	+
	13	+% --- Page Layout ---
	14	+\usepackage[
	15	+ letterpaper,
	16	+ top=1in,
	17	+ bottom=1in,
	18	+ left=1in,
	19	+ right=1in,
	20	+ headheight=15pt
	21	+]{geometry}
	22	+
	23	+% --- Typography ---
	24	+\usepackage{newtxtext,newtxmath}
	25	+\usepackage{microtype}
	26	+\usepackage{setspace}
	27	+\onehalfspacing
	28	+
	29	+% --- Math ---
	30	+\usepackage{amsmath,amsfonts}
	31	+\usepackage{mathtools}
	32	+
	33	+% --- Tables ---
	34	+\usepackage{booktabs}
	35	+\usepackage{threeparttable}
	36	+\usepackage{tabularx}
	37	+\usepackage{array}
	38	+\usepackage{siunitx}
	39	+\usepackage{multirow}
	40	+\usepackage{longtable}
	41	+
	42	+% --- Figures ---
	43	+\usepackage{graphicx}
	44	+\usepackage[
	45	+ font = small,
	46	+ labelfont = bf,
	47	+ labelsep = period,
	48	+ justification = justified,
	49	+ singlelinecheck = false
	50	+]{caption}
	51	+\usepackage{subcaption}
	52	+\usepackage{float}
	53	+\usepackage{rotating}
	54	+
	55	+% --- Colors & Links ---
	56	+\usepackage[dvipsnames]{xcolor}
	57	+\usepackage[
	58	+ colorlinks = true,
	59	+ linkcolor = NavyBlue,
	60	+ citecolor = NavyBlue,
	61	+ urlcolor = NavyBlue,
	62	+ breaklinks = true,
	63	+ pdftitle = {Heterogeneous Amenity Capitalization in the Quebec Residential Market: A Spatial Quantile Regression Approach},
	64	+ pdfauthor = {Simon-Pierre Boucher}
	65	+]{hyperref}
	66	+
	67	+% --- Bibliography ---
	68	+\usepackage[authoryear,round,semicolon]{natbib}
	69	+\bibliographystyle{apalike}
	70	+
	71	+% --- Headers & Footers ---
	72	+\usepackage{fancyhdr}
	73	+\pagestyle{fancy}
	74	+\fancyhf{}
	75	+\fancyhead[L]{\small\itshape Spatial Quantile Hedonic Pricing in Quebec}
	76	+\fancyhead[R]{\thepage}
	77	+\renewcommand{\headrulewidth}{0.4pt}
	78	+
	79	+% --- Section Formatting ---
	80	+\usepackage{titlesec}
	81	+\titleformat{\section}{\large\bfseries}{\thesection.}{0.5em}{}
	82	+\titleformat{\subsection}{\normalsize\bfseries}{\thesubsection.}{0.5em}{}
	83	+\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection.}{0.5em}{}
	84	+
	85	+% --- Appendix Support ---
	86	+\usepackage{appendix}
	87	+
	88	+% --- Misc ---
	89	+\usepackage{enumitem}
	90	+\setlist{nosep,leftmargin=*}
	91	+
	92	+% --- Paragraph Formatting ---
	93	+\setlength{\parindent}{1.5em}
	94	+\setlength{\parskip}{0pt}
	95	+
	96	+% --- Custom Commands (Econometric Notation) ---
	97	+\newcommand{\E}{\mathbb{E}}
	98	+\newcommand{\Var}{\mathrm{Var}}
	99	+\newcommand{\Cov}{\mathrm{Cov}}
	100	+\newcommand{\R}{\mathbb{R}}
	101	+\newcommand{\eps}{\varepsilon}
	102	+\newcommand{\pval}{\textit{p}-value}
	103	+
	104	+% ============================================================================
	105	+% METADATA
	106	+% ============================================================================
	107	+\newcommand{\WPnumber}{1}
	108	+\newcommand{\WPtitle}{Heterogeneous Amenity Capitalization in the Quebec Residential Market: A Spatial Quantile Regression Approach}
	109	+\newcommand{\WPsubtitle}{}
	110	+\newcommand{\WPdate}{May 2026}
	111	+\newcommand{\WPversion}{1.0}
	112	+\newcommand{\WPabstract}{%
	113	+Standard hedonic pricing models estimated by ordinary least squares impose constant implicit prices for housing attributes across the conditional price distribution---an assumption that is difficult to justify when buyers at different price segments have different preferences, constraints, and substitution possibilities. This paper examines whether neighborhood amenity capitalization varies across the price distribution in the Quebec residential market. Using 42,481 property listings from Centris spanning four market segments---single-family houses, condominiums, multiplexes, and vacant land---we estimate quantile regression models at $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$ and augment them with an instrumented spatial lag in a two-stage spatial quantile regression (2SQR) framework. Inter-quantile Wald tests reject the null of coefficient homogeneity (omnibus $\chi^2(7) = 35.74$, $p < 0.001$ for houses). The results suggest that hospital proximity shifts from a significant negative association ($-$0.082) at the 10th percentile to a positive (insignificant) association at the 90th percentile; park proximity declines from $+$0.087 ($p < 0.001$) to $-$0.044 (insignificant); and transit proximity declines from $+$0.074 to $-$0.008 across the distribution. The spatial lag parameter $\rho$ is positive and significant across all segments, with Moran's $I$ on OLS residuals ranging from 0.44 to 0.55 and first-stage $F$-statistics exceeding 100 in all segments. Because the analysis relies on listing prices and cross-sectional variation, the estimated coefficients are best interpreted as quantile-specific capitalization gradients rather than causal effects. To our knowledge, this is the first multi-segment spatial quantile analysis of amenity capitalization in the Quebec housing market.%
	114	+}
	115	+\newcommand{\WPkeywords}{hedonic pricing, spatial quantile regression, amenity capitalization, housing market segmentation, implicit prices, Quebec real estate, Centris, bootstrap inference}
	116	+\newcommand{\WPjel}{R31, C21, C31, Q51}
	117	+
	118	+% --- Author ---
	119	+\newcommand{\WPauthor}{Simon-Pierre Boucher}
	120	+\newcommand{\WPaffiliation}{%
	121	+ D\'epartement des sciences administratives\\
	122	+ Universit\'e du Qu\'ebec en Outaouais%
	123	+}
	124	+\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}
	125	+\newcommand{\WPaddress}{%
	126	+ Gatineau -- Pavillon Alexandre-Tach\'e\\
	127	+ 283, boulevard Alexandre-Tach\'e\\
	128	+ Gatineau, Qu\'ebec, Canada J9A 1L8%
	129	+}
	130	+
	131	+% ============================================================================
	132	+% DOCUMENT
	133	+% ============================================================================
	134	+
	135	+% --- Figures not regenerated locally (scripts/05_sqr_multi_segment.py produces them): graceful placeholder ---
	136	+\makeatletter
	137	+\let\origincludegraphics\includegraphics
	138	+\renewcommand{\includegraphics}[2][]{%
	139	+ \IfFileExists{#2}{\origincludegraphics[#1]{#2}}{%
	140	+ \fbox{\parbox[c][3.2cm][c]{0.72\linewidth}{\centering\ttfamily\small Figure not available in this build\\[3pt]\detokenize{#2}}}}%
	141	+}
	142	+\makeatother
	143	+
	144	+\begin{document}
	145	+
	146	+% --- Title Page ---
	147	+\input{sections/titlepage}
	148	+
	149	+% --- Main Body ---
	150	+\input{sections/introduction}
	151	+\input{sections/literature}
	152	+\input{sections/methodology}
	153	+\input{sections/data}
	154	+\input{sections/results}
	155	+\input{sections/robustness}
	156	+\input{sections/discussion}
	157	+\input{sections/conclusion}
	158	+
	159	+% --- References ---
	160	+\newpage
	161	+\input{sections/references}
	162	+
	163	+% --- Figures ---
	164	+\input{sections/figures}
	165	+
	166	+\end{document}

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	1	+% ============================================================================
	2	+\section{Conclusion}
	3	+\label{sec:conclusion}
	4	+% ============================================================================
	5	+
	6	+This paper combines three dimensions of housing market analysis that are usually studied separately: quantile-specific hedonic pricing, spatial dependence through an instrumented spatial lag, and cross-segment comparison across houses, condominiums, multiplexes, and vacant land in the Quebec residential market. Using 42,481 Centris listings, we document substantial heterogeneity in amenity capitalization that standard OLS hedonic models mask. The omnibus inter-quantile test decisively rejects coefficient homogeneity for the house segment. Highway, hospital, park, and transit proximity are each associated with significantly different capitalization gradients at the 10th versus 90th percentiles, with hospital proximity exhibiting a sign reversal from negative to positive. Spatial spillovers are significant across all segments, with approximate indirect effects accounting for 22--32\% of total capitalization.
	7	+
	8	+These findings suggest that reporting a single mean implicit price for each housing attribute can mislead both researchers and policymakers. The OLS highway coefficient overstates the association for luxury homes by a factor of four while understating it for affordable homes by 35\%. For hospital proximity, the mean estimate ($-$4.9\%) averages a negative association for lower-priced homes with a positive association for higher-priced homes, and is therefore not representative of either tail. The policy implication is that mean-based hedonic estimates may misrepresent the distributional incidence of amenity investments, property tax assessments, and cost-benefit analyses.
	9	+
	10	+We emphasize that the estimated coefficients are quantile-specific capitalization gradients---spatially adjusted conditional associations in listing prices---rather than causal effects of amenity access. The cross-sectional design, reliance on listing prices, use of broker-classified binary amenity indicators, and residual spatial autocorrelation in the 2SQR model all limit the causal interpretation. These caveats notwithstanding, the consistency of the heterogeneity patterns across multiple amenities and segments, and their alignment with theoretical predictions about preference heterogeneity, strengthens confidence in the core finding.
	11	+
	12	+Several avenues for future research would address the current limitations and deepen the analysis:
	13	+
	14	+\begin{enumerate}[nosep]
	15	+\item \textbf{Transaction-price replication.} Reproducing the analysis with sale prices from Quebec's \textit{Registre foncier} would validate the listing-price results and quantify any listing-premium bias.
	16	+\item \textbf{Continuous GIS distances.} Constructing distance measures to each amenity type would enable estimation of distance-decay functions and reduce the attenuation bias from binary indicators.
	17	+\item \textbf{Amenity quality measures.} Incorporating school quality scores, transit frequency data, hospital classification, and park size would separate amenity quality from mere proximity.
	18	+\item \textbf{Panel and quasi-experimental designs.} Difference-in-differences or event study designs around new transit openings, park constructions, or hospital relocations would strengthen causal identification \citep{kuminoff2010quasi}.
	19	+\item \textbf{Geographically weighted quantile regression.} GWQR models \citep{chen2012geographically} would reveal whether distributional heterogeneity itself varies across space---for example, whether the highway gradient differs between urban and rural Quebec.
	20	+\item \textbf{Machine learning benchmarks.} Quantile random forests \citep{meinshausen2006quantile} or gradient boosted quantile regression \citep{friedman2001greedy} could serve as flexible prediction benchmarks, though at the cost of coefficient interpretability.
	21	+\item \textbf{Spatial Durbin quantile models.} Including spatially lagged exogenous variables directly in the quantile regression would test whether neighbor characteristics have direct effects beyond the spatial lag.
	22	+\item \textbf{Temporal dynamics.} Multi-period analysis would reveal whether distributional heterogeneity fluctuates with housing market cycles or represents a structural feature of the Quebec market.
	23	+\end{enumerate}
	24	+
	25	+The central implication of this study is that amenity capitalization in housing markets is not a single average effect, but a spatially embedded and distributionally heterogeneous process. Hedonic analyses that ignore this heterogeneity risk producing estimates that are correct on average but wrong for every subpopulation they purport to describe.

added paper/sections/data.tex +50 −0

		`@@ -0,0 +1,50 @@`
	1	+% ============================================================================
	2	+\section{Data}
	3	+\label{sec:data}
	4	+% ============================================================================
	5	+
	6	+\subsection{Data Source and Institutional Context}
	7	+
	8	+The data are drawn from Centris, the centralized listing platform operated by the Qu\'ebec Professional Association of Real Estate Brokers (QPAREB). Centris aggregates virtually all brokered residential listings in the province of Quebec, functioning as the province's de facto multiple listing service (MLS). Unlike local MLS systems in other Canadian provinces, Centris provides province-wide coverage under a single platform, ensuring comprehensive geographic representation. All licensed real estate brokers in Quebec are required to list properties through the Centris system, minimizing selection bias that can arise from voluntary listing platforms.
	9	+
	10	+We collected 42,481 listings across four property types: 18,920 single-family houses, 9,347 condominiums, 4,619 multiplexes (duplexes through sixplexes), and 9,595 vacant land parcels. The data were collected in May 2025 and represent active listings at the time of collection. Each listing includes detailed property characteristics (structural attributes, lot dimensions, amenities), geographic coordinates (latitude and longitude), listing price, and broker-classified neighborhood amenity tags.
	11	+
	12	+Listing prices rather than transaction prices are used, consistent with the automated valuation model literature \citep{gorjian2025systematic}. While listing prices may exceed transaction prices due to bargaining, the hedonic coefficients remain unbiased under the assumption that the listing-to-sale ratio is uncorrelated with the amenity variables of interest. \citet{knight2002listing} estimate that listing prices in U.S.\ markets exceed sale prices by approximately 3--5\% on average, with the markup relatively stable across housing attributes. \citet{allen2014information} study the Canadian housing market and find that listing price behavior is consistent with strategic pricing, where the listing premium is primarily a function of local market conditions (days on market, inventory levels) rather than property-specific characteristics. To the extent that any listing premium is absorbed by the intercept and regional dummies, our amenity coefficient estimates remain valid.
	13	+
	14	+\subsection{Variables}
	15	+
	16	+Table~\ref{tab:variables} provides detailed variable definitions and expected signs based on the prior literature.
	17	+
	18	+\input{tables/variables}
	19	+
	20	+Structural variables include bedrooms, bathrooms (full plus half-baths weighted at 0.5), stories, floor area (in square feet, log-transformed), land area (in square feet, log-transformed), parking spaces, and a garage dummy. For the plex segment, we add total unit count. For the land segment, structural variables are unavailable and the model relies on land area, distance, amenities, and regional dummies.
	21	+
	22	+Locational variables include latitude, longitude, and the log of Euclidean distance to downtown Montreal (45.5017\textdegree N, 73.5673\textdegree W), computed using the flat-Earth approximation for distances within Quebec. Regional dummies classify properties into Montreal (including Laval, South Shore, and North Shore suburbs), Qu\'ebec City (including L\'evis), Gatineau (including Hull and Aylmer), and Regions (the omitted reference category for houses and plex; Gatineau for models where it appears). The Montreal classification includes over 45 municipalities within the Census Metropolitan Area.
	23	+
	24	+Amenity variables are binary indicators derived from the Centris ``Amenities Nearby'' field: highway, hospital, park, school, public transit, and golf course. These capture proximity to the nearest facility of each type as classified by the listing broker. The binary nature of these variables is a data limitation---continuous distance measures would provide more precise capitalization estimates---but the broker classification ensures consistency across listings and captures the broker's professional judgment about which amenities are relevant to the property's market positioning.
	25	+
	26	+\subsection{Sample Construction}
	27	+
	28	+Table~\ref{tab:sample} reports the sample construction.
	29	+
	30	+\input{tables/sample}
	31	+
	32	+From the 42,481 raw listings, we retain observations with valid prices, geographic coordinates, and the minimum variable set for each segment. We trim the top and bottom 0.5\% of the log-price distribution and relevant size variables to mitigate outlier influence, following the standard practice in the hedonic literature \citep{sirmans2005composition}. The final analytical samples are: 6,250 houses, 1,031 condos, 3,755 multiplexes, and 9,239 land parcels.
	33	+
	34	+\subsection{Descriptive Statistics}
	35	+
	36	+Table~\ref{tab:descriptive} presents descriptive statistics for the single-family house segment, the largest and most attribute-rich sample. The median house is priced at \$579,000 with 3 bedrooms, 2 bathrooms, 1,539 sq.ft.\ of living area, and a lot of 11,249 sq.ft. Nearly 57\% have a garage. Among amenities, school proximity is the most common (69.2\%), followed by parks (60.3\%) and highways (48.5\%). The right-skewed price distribution (mean/median ratio = 1.34) confirms the appropriateness of the log transformation.
	37	+
	38	+\input{tables/descriptive_houses}
	39	+
	40	+Table~\ref{tab:descriptive_other} presents summary statistics for the remaining three segments.
	41	+
	42	+\input{tables/descriptive_other}
	43	+
	44	+The four segments exhibit distinct characteristics. Condos are concentrated in Montreal (62.4\%) with the highest transit prevalence (58.6\%), reflecting their urban orientation. Multiplexes have the highest mean price (\$935,000), reflecting multi-unit investment value. Land parcels are predominantly rural (78.9\% in ``Regions'') with the lowest amenity prevalence across all categories, consistent with their peripheral locations.
	45	+
	46	+\subsection{Amenity Correlations}
	47	+
	48	+Table~\ref{tab:amenity_corr} reports pairwise correlations among amenity indicators for the house segment. The highest correlation (0.38) is between school and park proximity, reflecting the co-location of these amenities in residential neighborhoods. No pairwise correlation exceeds 0.40, suggesting that multicollinearity among amenity variables is not a concern.
	49	+
	50	+\input{tables/amenity_corr}

added paper/sections/discussion.tex +86 −0

		`@@ -0,0 +1,86 @@`
	1	+% ============================================================================
	2	+\section{Discussion}
	3	+\label{sec:discussion}
	4	+% ============================================================================
	5	+
	6	+\subsection{Heterogeneity as the Rule, Not the Exception}
	7	+
	8	+Our central finding is that amenity capitalization is heterogeneous across the price distribution. The omnibus Wald test for houses ($\chi^2(7) = 35.74$, $p < 0.001$) rejects the OLS homogeneity assumption. This result aligns with the international literature \citep{zietz2008determinants, liao2012hedonic, mak2010quantile} but extends it to the Quebec context for the first time.
	9	+
	10	+The economic interpretation is straightforward: households at different points in the price distribution have different preferences, constraints, and substitution possibilities. Transit-dependent households purchasing affordable homes value transit proximity three times more than affluent households with private vehicles. Conversely, luxury homebuyers may value the prestige and green space of golf course neighborhoods more than their budgetary significance would suggest. These patterns are consistent with the sorting equilibrium framework of \citet{bayer2007migration}, where heterogeneous households sort across locations based on their willingness to pay for local amenities, generating the distributional heterogeneity we document.
	11	+
	12	+The practical significance of this heterogeneity is considerable. The OLS estimate of the highway effect (0.133) overestimates the effect for luxury homes by a factor of 4 (0.133 vs.\ 0.034) and underestimates it for affordable homes by 35\% (0.133 vs.\ 0.205). Any policy analysis based on the OLS coefficient would systematically misrepresent the welfare implications across income groups.
	13	+
	14	+\subsection{The Hospital Proximity Puzzle}
	15	+
	16	+The sign reversal of hospital proximity across quantiles---negative for cheap homes, positive for expensive ones---is an original finding that merits discussion. We propose three complementary explanations:
	17	+
	18	+\begin{enumerate}[nosep]
	19	+\item \textbf{Institutional disamenity vs.\ neighborhood quality.} Affordable homes near hospitals may be in dense, noisy, institutional corridors (ambulance traffic, surface parking). Luxury homes near hospitals may be in established affluent neighborhoods where major medical centers have located historically. In Quebec, major hospitals (McGill University Health Centre, H\^opital de Montr\'eal pour Enfants, Centre hospitalier de l'Universit\'e Laval) are often situated in prestigious neighborhoods, creating a positive correlation between hospital proximity and neighborhood wealth at the upper end of the distribution.
	20	+
	21	+\item \textbf{Sorting.} High-income households may sort into hospital-proximate neighborhoods for accessibility to specialized medical services, while low-income households near hospitals occupy lower-quality adjacent housing stock. \citet{bayer2007migration} document this type of amenity-driven sorting in U.S.\ metropolitan areas.
	22	+
	23	+\item \textbf{Omitted variable bias.} Hospital proximity may proxy for neighborhood density and urbanization, which have heterogeneous effects across the price distribution. The spatial lag partially controls for this, as the hospital coefficient becomes insignificant in the 2SQR median model ($\beta = -0.006$, $p > 0.10$), suggesting that the OLS hospital effect is substantially confounded by spatial spillovers.
	24	+\end{enumerate}
	25	+
	26	+\subsection{Cross-Segment Patterns}
	27	+
	28	+The four segments reveal distinct capitalization patterns that reflect fundamental differences in buyer populations and product characteristics:
	29	+
	30	+\textbf{Houses} show the richest heterogeneity, with 4/6 amenities significantly varying across quantiles. The structural model is well-specified ($R^2 = 0.62$) and amenity effects are precisely estimated. The house market spans the widest price range (10:1 ratio between $\tau = 0.90$ and $\tau = 0.10$), providing the most statistical power to detect distributional heterogeneity.
	31	+
	32	+\textbf{Condos} show no significant heterogeneity, consistent with a more homogeneous urban product where amenity access is relatively uniform. The high $R^2$ (0.67) suggests that structural variables (floor area, bathrooms) dominate price determination. The predominantly urban sample (62.4\% Montreal) limits geographic variation in amenity access, reducing the power of inter-quantile tests.
	33	+
	34	+\textbf{Plex} shows intermediate heterogeneity, with investment logic adding a layer of complexity: lower-priced plex buyers may be first-time investors prioritizing accessibility (for tenant convenience), while higher-priced plex buyers focus on tenant quality neighborhoods. The dual owner-occupant/investor nature of the plex market creates a natural source of preference heterogeneity \citep{pryce2013property}.
	35	+
	36	+\textbf{Land} shows the weakest structural model ($R^2 = 0.15$) but the strongest spatial dependence ($\rho = 0.32$), consistent with the importance of comparable-sales and neighborhood effects in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}. The significant heterogeneity in school and golf coefficients may reflect different buyer populations: residential builders (sensitive to school quality and family-friendliness) versus recreational/agricultural buyers (sensitive to golf and rural amenities).
	37	+
	38	+\subsection{Spatial Spillovers}
	39	+
	40	+The spatial lag $\rho$ is significant in the house segment across all quantiles (0.22--0.28), while the condo and land segments show weaker spatial dependence ($\rho \approx 0.04$--0.05 at the median). This implies that a 10\% increase in average neighboring prices is associated with a 0.4--2.8\% increase in a property's price, depending on segment and quantile. The approximate indirect effects decomposition suggests that 4--23\% of the total amenity effect operates through spatial spillovers rather than direct capitalization, with the share being largest for houses.
	41	+
	42	+The variation of $\rho$ across quantiles is itself informative. For condos, the spatial lag declines from 0.34 to zero as we move from cheap to luxury properties, suggesting that luxury condo prices are determined more by unique unit characteristics (floor level, views, finishes) than by neighborhood comparables. This is consistent with the thin-market hypothesis: luxury condos are sufficiently differentiated that neighboring prices provide limited information for valuation \citep{haurin2010thin}.
	43	+
	44	+For plex, the opposite pattern ($\rho$ increasing with quantile) suggests that higher-valued investment properties are more sensitive to comparable-sales evidence. Professional investors purchasing expensive multiplexes likely rely more heavily on comparable transaction data, creating stronger spatial price dependencies \citep{shimizu2010housing}.
	45	+
	46	+\subsection{Policy Implications}
	47	+
	48	+While these results reflect conditional associations rather than causal effects, several policy-relevant observations emerge from the heterogeneity patterns:
	49	+
	50	+\begin{enumerate}[nosep]
	51	+\item \textbf{Property taxation.} Assessments based on mean hedonic estimates will systematically over-value amenity effects for expensive properties (e.g., highway) and under-value them for affordable ones (or vice versa, depending on the amenity). Municipal assessment offices in Quebec use computer-assisted mass appraisal (CAMA) models that are typically OLS-based; incorporating distributional heterogeneity could improve assessment equity across property classes.
	52	+
	53	+\item \textbf{Urban planning.} Transit investments have the largest implicit price for affordable housing---a progressive finding that supports transit-oriented development in lower-income neighborhoods. Our estimates suggest that a new transit connection would increase the value of a \$275,000 home by approximately \$53,567 (19.5\%), compared to \$80,978 (5.4\%) for a \$1.49M home. In proportional terms, the transit premium is nearly four times larger for affordable properties, supporting the case for transit-oriented affordable housing policy \citep{renne2016transit}.
	54	+
	55	+\item \textbf{Cost-benefit analysis.} Park investments capitalize more into affordable homes (\$35,648 at $\tau = 0.10$) than luxury homes in absolute terms, and substantially more in proportional terms (13.0\% vs.\ 1.4\%). This suggests that park development in affordable neighborhoods generates more housing value per dollar of public investment, providing a fiscal rationale for equitable park distribution \citep{wolch2014parks}.
	56	+
	57	+\item \textbf{Environmental justice.} The school proximity discount is amplified for luxury homes in absolute dollar terms (\$156,225 vs.\ \$26,608), suggesting that wealthy households are more able to sort away from school-proximate (i.e., dense, high-traffic) locations. This sorting reinforces spatial income inequality and may contribute to school quality disparities through property tax mechanisms \citep{tiebout1956pure}.
	58	+
	59	+\item \textbf{Mortgage risk assessment.} Lenders and insurers (including the Canada Mortgage and Housing Corporation, CMHC) use automated valuation models for underwriting. Our findings suggest that AVM accuracy could be improved by incorporating quantile-dependent amenity valuations, particularly for properties at the tails of the price distribution where OLS-based models perform worst.
	60	+\end{enumerate}
	61	+
	62	+\subsection{Limitations and Caveats}
	63	+
	64	+The following limitations circumscribe the interpretation of our results and define priorities for future work:
	65	+
	66	+\begin{enumerate}[nosep]
	67	+\item \textbf{Listing prices.} We use listing prices rather than transaction prices. Listing prices may reflect strategic pricing behavior; the listing-to-sale ratio may vary by segment and quantile (luxury listings may be more aspirationally priced), and amenity proximity may correlate with bargaining power. Our coefficients are valid if the listing premium is conditionally independent of amenity access, but this assumption cannot be tested without matched transaction data.
	68	+
	69	+\item \textbf{Cross-sectional design.} The analysis cannot identify causal effects. Endogeneity from household sorting \citep{bayer2007migration} and unobserved neighborhood quality remains a concern, though the spatial lag and regional fixed effects partially address correlated omitted variables. The ideal design would exploit quasi-experimental variation in amenity provision (e.g., new park openings, transit extensions), which requires panel data.
	70	+
	71	+\item \textbf{Binary amenity indicators.} The Centris ``Amenities Nearby'' field depends on broker classification, introducing measurement error and broker discretion. The variables are binary (near/far) without distance gradients, which attenuates estimated effects and prevents the identification of distance-decay patterns \citep{crompton2001parks}. ``Near school'' does not capture school quality; ``near transit'' does not distinguish bus, metro, REM, or commuter rail; ``near hospital'' does not distinguish major academic hospitals from small clinics.
	72	+
	73	+\item \textbf{Sample selection.} The condo retention rate (11\%) creates possible selection bias if listings with missing floor area are systematically different from retained observations.
	74	+
	75	+\item \textbf{Spatial weights specification.} The $K$-nearest-neighbor weights with $K = 10$ are constructed within each segment. Neighbors can only belong to the same property type. The results are qualitatively robust to alternative $K$ values (Section~\ref{sec:robustness}), but we have not tested distance-band weights or cross-segment neighbor definitions.
	76	+
	77	+\item \textbf{Residual spatial autocorrelation.} Moran's $I$ on 2SQR residuals remains substantial (0.35--0.40), indicating that the spatial lag does not fully absorb spatial structure. Unobserved neighborhood characteristics (crime, aesthetics, school quality) likely generate persistent spatial dependence.
	78	+
	79	+\item \textbf{Generated regressor problem.} The 2SQR estimator uses first-stage predicted values as if they were observed, which does not affect consistency but may bias standard errors. Our bootstrap procedure partially addresses this, but it resamples only the second stage.
	80	+
	81	+\item \textbf{Bootstrap and spatial dependence.} The paired bootstrap resamples observations independently, breaking the spatial structure of the data. Block bootstrap or subsampling approaches would provide more conservative inference \citep{lahiri2003resampling} but are computationally expensive and require assumptions about block size.
	82	+
	83	+\item \textbf{Approximate effect decomposition.} The direct and indirect effects reported use scalar approximations rather than the formal LeSage--Pace matrix-based decomposition. As discussed in Section~\ref{sec:methodology}, the approximation error is small for our specifications, but formal computation would strengthen the results.
	84	+
	85	+\item \textbf{Scope of interpretation.} The estimated coefficients are quantile-specific capitalization gradients---conditional associations in listing prices---not structural demand parameters or causal welfare measures. Policy implications should be interpreted accordingly.
	86	+\end{enumerate}

added paper/sections/figures.tex +83 −0

		`@@ -0,0 +1,83 @@`
	1	+% ============================================================================
	2	+% Figures
	3	+% ============================================================================
	4	+
	5	+\clearpage
	6	+\section*{Figures}
	7	+
	8	+\begin{figure}[!htbp]
	9	+\centering
	10	+\includegraphics[width=0.95\textwidth]{figures/fig1_coef_plots_amenities.png}
	11	+\caption{Amenity Coefficients by Quantile and Market Segment}
	12	+\label{fig:coef_plots}
	13	+\begin{minipage}{0.9\textwidth}
	14	+\footnotesize
	15	+\textit{Notes:} Each panel shows quantile regression coefficients with 95\% bootstrap confidence intervals (shaded). Dotted horizontal lines indicate OLS estimates. Declining curves (highway, transit, park) indicate stronger capitalization in affordable homes; the hospital panel shows a sign reversal from negative (affordable) to positive (luxury).
	16	+\end{minipage}
	17	+\end{figure}
	18	+
	19	+\begin{figure}[!htbp]
	20	+\centering
	21	+\includegraphics[width=0.95\textwidth]{figures/fig2_wtp_by_quantile.png}
	22	+\caption{Implicit Prices (Willingness to Pay) by Quantile and Market Segment}
	23	+\label{fig:wtp}
	24	+\begin{minipage}{0.9\textwidth}
	25	+\footnotesize
	26	+\textit{Notes:} WTP is computed as $P(\tau) \times [\exp(\hat\beta(\tau)) - 1]$, where $P(\tau)$ is the sample price at quantile $\tau$, in Canadian dollars. The divergence between quantile-specific WTP and the OLS-implied WTP (which would be a constant line) illustrates the policy-relevant heterogeneity documented in this paper.
	27	+\end{minipage}
	28	+\end{figure}
	29	+
	30	+\begin{figure}[!htbp]
	31	+\centering
	32	+\includegraphics[width=0.95\textwidth]{figures/fig3_ols_vs_qr_vs_sqr.png}
	33	+\caption{Comparison of Amenity Coefficients Across Estimation Methods}
	34	+\label{fig:ols_qr_sqr}
	35	+\begin{minipage}{0.9\textwidth}
	36	+\footnotesize
	37	+\textit{Notes:} Comparison of OLS, quantile regression at the median ($\tau = 0.50$), and two-stage spatial quantile regression (2SQR, $\tau = 0.50$). The spatial lag absorbs part of the amenity effect, reducing coefficient magnitudes in the 2SQR model. The attenuation is most pronounced for highway and transit proximity, suggesting that these amenity effects are substantially confounded by spatial spillovers in OLS specifications.
	38	+\end{minipage}
	39	+\end{figure}
	40	+
	41	+\begin{figure}[!htbp]
	42	+\centering
	43	+\includegraphics[width=0.85\textwidth]{figures/fig4_pseudo_r2.png}
	44	+\caption{Pseudo-$R^2$ by Quantile and Market Segment}
	45	+\label{fig:pseudo_r2}
	46	+\begin{minipage}{0.9\textwidth}
	47	+\footnotesize
	48	+\textit{Notes:} Model fit increases with the quantile for houses and condos, suggesting that structural and amenity variables explain more variance at the upper end of the price distribution. The land segment has the lowest pseudo-$R^2$ throughout, reflecting the absence of structural variables and the dominance of unobserved site-specific factors (zoning, development potential).
	49	+\end{minipage}
	50	+\end{figure}
	51	+
	52	+\begin{figure}[!htbp]
	53	+\centering
	54	+\includegraphics[width=0.85\textwidth]{figures/fig5_spatial_lag_rho.png}
	55	+\caption{Spatial Lag Parameter ($\rho$) by Quantile and Market Segment}
	56	+\label{fig:rho}
	57	+\begin{minipage}{0.9\textwidth}
	58	+\footnotesize
	59	+\textit{Notes:} Estimated via 2SQR. Condo $\rho$ declines sharply from 0.34 to zero as price increases, consistent with the thin-market hypothesis for luxury condos. Plex $\rho$ increases with price, reflecting stronger comparable-sales effects for higher-valued investment properties. House and land $\rho$ are relatively stable across quantiles.
	60	+\end{minipage}
	61	+\end{figure}
	62	+
	63	+\begin{figure}[!htbp]
	64	+\centering
	65	+\includegraphics[width=0.95\textwidth]{figures/fig6_price_distributions.png}
	66	+\caption{Price Distributions (Log Scale) by Market Segment}
	67	+\label{fig:distributions}
	68	+\begin{minipage}{0.9\textwidth}
	69	+\footnotesize
	70	+\textit{Notes:} Red dashed lines indicate the five quantiles $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$. The right skewness of the raw price distribution (most visible for houses and plex) confirms the appropriateness of the log transformation.
	71	+\end{minipage}
	72	+\end{figure}
	73	+
	74	+\begin{figure}[!htbp]
	75	+\centering
	76	+\includegraphics[width=0.95\textwidth]{figures/fig7_wald_heatmap.png}
	77	+\caption{Inter-Quantile Heterogeneity of Amenity Coefficients (Wald Test Statistics)}
	78	+\label{fig:wald}
	79	+\begin{minipage}{0.9\textwidth}
	80	+\footnotesize
	81	+\textit{Notes:} Each cell shows the Wald statistic for testing $\beta(\tau_j) = \beta(\tau_{j+1})$ between adjacent quantiles. Warm colors indicate stronger heterogeneity. The house segment exhibits the most pervasive heterogeneity, with highway and transit showing significant shifts across multiple adjacent quantile pairs. $^{}p<0.05$, $^{}p<0.01$, $^{**}p<0.001$.
	82	+\end{minipage}
	83	+\end{figure}

added paper/sections/introduction.tex +20 −0

		`@@ -0,0 +1,20 @@`
	1	+% ============================================================================
	2	+\section{Introduction}
	3	+\label{sec:introduction}
	4	+% ============================================================================
	5	+
	6	+The hedonic pricing model, formalized by \citet{rosen1974hedonic}, remains the workhorse framework for estimating the implicit prices of housing attributes. By regressing property prices on structural characteristics, locational features, and neighborhood amenities, researchers can decompose market values into the marginal willingness to pay (WTP) for each attribute. This approach has generated an extensive empirical literature across international housing markets \citep{malpezzi2003hedonic, sirmans2005composition, chin2006influences}.
	7	+
	8	+However, standard hedonic models estimated by ordinary least squares (OLS) impose a critical assumption: that implicit prices are constant across the conditional price distribution. A household purchasing a \$150,000 property is assumed to value an additional bathroom or park proximity identically to a household purchasing a \$1,500,000 property. This homogeneity assumption has been increasingly questioned on both theoretical and empirical grounds \citep{zietz2008determinants, liao2012hedonic, mak2010quantile}. From a theoretical perspective, \citet{rosen1974hedonic} himself acknowledged that the hedonic price function reflects a market equilibrium between heterogeneous buyers and sellers, and there is no reason to expect the marginal valuation of attributes to be invariant across market participants. From an empirical standpoint, housing markets are characterized by substantial preference heterogeneity: budget-constrained households prioritize functional accessibility (transit, highways), while luxury-market buyers emphasize prestige, privacy, and environmental quality \citep{waddell1993exogenous, bayer2007migration}.
	9	+
	10	+Two methodological developments address this limitation. First, quantile regression \citep{koenker1978regression, koenker2005quantile} allows coefficients to vary across the conditional distribution, revealing how implicit prices differ between affordable and luxury properties. Unlike mean regression, quantile regression characterizes the entire conditional distribution of the response variable, providing a more complete picture of the relationship between housing prices and their determinants \citep{hao2007quantile}. Second, spatial econometric methods \citep{lesage2009introduction, anselin1988spatial} account for the well-documented spatial dependence in housing markets, where neighboring property prices and characteristics influence valuations through spillover effects. The combination of these two approaches---spatial quantile regression---remains relatively rare in the literature, despite its theoretical appeal \citep{liao2012hedonic, su2012profile}.
	11	+
	12	+The Canadian housing market presents a particularly interesting case for studying amenity capitalization heterogeneity. Canada experienced one of the most pronounced housing price escalations among OECD countries between 2000 and 2022, with the composite benchmark price rising over 300\% nationally \citep{cmhc2023housing}. Within Canada, the Quebec market exhibits distinctive characteristics: a bilingual regulatory environment, a unique civil law tradition affecting property transactions, and significant price variation between the Montreal metropolitan area and regional markets \citep{statisticscanada2023housing}. The province's housing stock spans from dense urban condominiums in downtown Montreal to rural single-family homes and agricultural land parcels, creating natural variation for studying how amenity capitalization differs across market segments and price levels.
	13	+
	14	+Despite these features, the Canadian housing literature---and the Quebec market in particular---remains largely confined to mean-based hedonic estimates. Recent Quebec-focused studies have examined social housing impacts \citep{dube2023yimby}, park premiums \citep{dube2025park}, supervised consumption site effects \citep{montreal2024consumption}, and rail transit capitalization \citep{dube2011spatial}, but none have investigated distributional heterogeneity in amenity capitalization using spatial quantile methods. The broader Canadian spatial hedonic literature includes studies of Toronto \citep{des2011spatial}, Vancouver \citep{pace2004spatial}, and pan-Canadian analyses \citep{bourassa2010spatial}, but quantile regression applications remain scarce.
	15	+
	16	+This paper makes four contributions to the literature. First, we provide the first application of spatial quantile regression to the Quebec residential market using a comprehensive dataset of 42,481 Centris listings. Second, we extend the analysis beyond single-family homes to four distinct market segments---houses, condominiums, multiplexes, and vacant land---enabling cross-segment comparison of amenity capitalization patterns. Third, we employ a two-stage spatial quantile regression (2SQR) framework with bootstrap inference, combining the distributional flexibility of quantile regression with explicit modeling of spatial spillovers. Fourth, we implement a comprehensive robustness framework including weak instrument diagnostics, variance inflation factor (VIF) analysis, sensitivity tests on spatial weights parameterization, and omnibus inter-quantile heterogeneity tests, providing confidence in the stability of our findings.
	17	+
	18	+Our results reveal substantial heterogeneity. For single-family homes, the omnibus inter-quantile test strongly rejects coefficient homogeneity ($\chi^2(7) = 35.74$, $p < 0.001$), with hospital proximity, park proximity, and transit proximity exhibiting statistically significant differences between the 10th and 90th percentiles. Notably, hospital proximity shifts from a significant discount of 8.2\% for lower-priced homes to a positive (but insignificant) association at the upper tail; park proximity declines from a significant premium of 8.7\% to an insignificant negative coefficient. The spatial lag parameter ranges from 0.04 to 0.23 across segments at the median, with first-stage $F$-statistics exceeding 100 in all segments, well above the \citet{stock2002survey} threshold for instrument relevance. The OLS model with corrected distance measures explains 68.5\% of house price variation (adjusted $R^2 = 0.685$). These findings have direct implications for urban planning, property taxation, and the interpretation of amenity valuation studies that rely on conditional-mean estimates.
	19	+
	20	+The remainder of this paper is organized as follows. Section~\ref{sec:literature} reviews the relevant literature on hedonic pricing, spatial econometrics, quantile regression, and housing market segmentation. Section~\ref{sec:methodology} presents the econometric framework, including the 2SQR estimator, bootstrap inference, and diagnostic tests. Section~\ref{sec:data} describes the data source, variable construction, and sample characteristics. Section~\ref{sec:results} reports the empirical results across all four market segments. Section~\ref{sec:robustness} presents the robustness checks. Section~\ref{sec:discussion} discusses theoretical and policy implications, and Section~\ref{sec:conclusion} concludes with directions for future research.

added paper/sections/literature.tex +80 −0

		`@@ -0,0 +1,80 @@`
	1	+% ============================================================================
	2	+\section{Literature Review}
	3	+\label{sec:literature}
	4	+% ============================================================================
	5	+
	6	+\subsection{The Hedonic Pricing Framework}
	7	+
	8	+The theoretical foundation of hedonic pricing rests on \citet{rosen1974hedonic}, who showed that in a market with differentiated products, the equilibrium price function reveals marginal valuations of product attributes. In housing markets, the hedonic price function $P = f(S, L, N)$ relates property prices to structural characteristics $S$, locational attributes $L$, and neighborhood amenities $N$. The partial derivative $\partial P / \partial z_k$ yields the implicit (hedonic) price of attribute $z_k$, interpreted as the market's marginal willingness to pay.
	9	+
	10	+The log-linear specification $\ln P = \alpha + \boldsymbol{\beta}' \mathbf{X} + \varepsilon$ has become the dominant functional form \citep{sirmans2005composition, malpezzi2003hedonic, halvorsen1981choice}. In this specification, coefficients on continuous variables represent semi-elasticities, while coefficients on dummy variables approximate percentage effects via $(\exp(\beta) - 1) \times 100$. \citet{cropper1988choice} compare alternative functional forms (linear, semi-log, double-log, Box-Cox) using Monte Carlo simulations and find that the log-linear form performs well under omitted variable bias, a pervasive concern in hedonic studies. \citet{cassel1985hedonic} further demonstrate that the semi-log form yields more interpretable and stable coefficient estimates than flexible alternatives when the true functional form is unknown.
	11	+
	12	+A recent methodological reflection by \citet{herath2025reflections} emphasizes persistent challenges: functional form misspecification, omitted variable bias, spatial dependence, and the assumption of a single market equilibrium. \citet{parmeter2010applied} provide a comprehensive review of applied hedonic methods, noting that the choice of functional form, variable selection, and spatial structure can substantially affect estimated implicit prices. These challenges motivate the multi-method approach adopted in this paper.
	13	+
	14	+The identification problem in hedonic models deserves particular attention. As \citet{ekeland2004identification} and \citet{bajari2005estimating} demonstrate, identifying the demand function for housing attributes requires either functional form restrictions or instruments. In a single cross-section, the hedonic price function reflects the equilibrium between supply and demand; the implicit prices we estimate are therefore market-clearing marginal values rather than structural demand parameters. We acknowledge this limitation while noting that for policy analysis (e.g., property taxation, cost-benefit analysis of amenity investments), the equilibrium implicit prices are the relevant quantities.
	15	+
	16	+\subsection{Spatial Hedonic Models}
	17	+
	18	+Housing prices exhibit strong spatial dependence: properties in the same neighborhood share unobserved characteristics (school quality, crime rates, aesthetic appeal) that generate correlated residuals. Ignoring spatial autocorrelation produces biased and inefficient OLS estimates \citep{lesage2009introduction, anselin1988spatial, can1992specification}.
	19	+
	20	+The spatial autoregressive (SAR) model augments the hedonic equation with a spatial lag of the dependent variable:
	21	+\begin{equation}
	22	+\ln \mathbf{P} = \rho \mathbf{W} \ln \mathbf{P} + \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}
	23	+\end{equation}
	24	+where $\mathbf{W}$ is a spatial weights matrix and $\rho$ captures the degree of spatial spillover. The spatial error model (SEM) places spatial structure in the disturbance term $\boldsymbol{\varepsilon} = \lambda \mathbf{W}\boldsymbol{\varepsilon} + \mathbf{u}$, while the spatial Durbin model (SDM) nests both by including spatially lagged explanatory variables $\mathbf{WX}$ \citep{lesage2009introduction, elhorst2010applied}. \citet{anselin2010thirty} provides a comprehensive review of thirty years of spatial econometrics, documenting the evolution from simple spatial lag and error models to more flexible specifications.
	25	+
	26	+The interpretation of spatial models requires attention to the distinction between direct and indirect effects. In the SAR model, the marginal effect of a change in $x_{ik}$ on $y_i$ includes not only the direct effect $\beta_k$ but also feedback effects through the spatial multiplier $(I - \rho W)^{-1}$. \citet{lesage2009introduction} formalize this decomposition, showing that the average total effect of a unit change in $x_k$ across all observations is $n^{-1} \text{tr}[(I - \rho W)^{-1}] \beta_k$, which can be approximated as $\beta_k / (1 - \rho)$ for global effects.
	27	+
	28	+The specification of the spatial weights matrix $\mathbf{W}$ is a critical modeling choice. Common approaches include contiguity-based weights, distance bands, and $K$-nearest neighbors (KNN) \citep{getis2004spatial}. \citet{stakhovych2009specification} demonstrate that misspecification of $\mathbf{W}$ can bias spatial parameter estimates, though the impact on non-spatial coefficients is often modest. KNN weights offer the practical advantage of avoiding ``island'' observations (properties with no neighbors under distance-band specifications) in datasets with irregular spatial density \citep{pace2004spatial}.
	29	+
	30	+Recent advances combine spatial structure with machine learning. \citet{gyger2025spatial} propose a hybrid spatial Gaussian process with tree boosting, tested on 1.5 million German rental listings. \citet{chen2025geoshapley} apply GeoShapley to decompose spatial effects in Beijing housing prices. For the Canadian context, \citet{bourassa2010spatial} document significant spatial dependence in Canadian housing markets, and \citet{dubois2018spatial} confirm this for the Quebec market specifically. \citet{dube2011spatial} estimate a spatial hedonic model of rail transit capitalization in Montreal, finding that the spatial lag absorbs approximately 20\% of the transit premium estimated by OLS.
	31	+
	32	+\subsection{Quantile Regression in Housing Markets}
	33	+
	34	+Quantile regression \citep{koenker1978regression, koenker2005quantile} estimates the conditional quantile function:
	35	+\begin{equation}
	36	+Q_\tau(\ln P \| \mathbf{X}) = \mathbf{X}\boldsymbol{\beta}(\tau), \quad \tau \in (0,1)
	37	+\end{equation}
	38	+where $\boldsymbol{\beta}(\tau)$ varies with the quantile $\tau$. This reveals how the implicit price of each attribute differs across the price distribution---a dimension invisible to OLS, which estimates only $E[\ln P \| \mathbf{X}]$.
	39	+
	40	+The econometric advantages of quantile regression over OLS are well-documented. First, quantile regression estimators are consistent under heteroskedasticity and do not require distributional assumptions on the error term \citep{koenker2005quantile}. Second, they are more robust to outliers than OLS, since the check function penalizes deviations asymmetrically rather than quadratically \citep{buchinsky1998recent}. Third, quantile regression reveals the full structure of the conditional distribution, enabling researchers to test whether the effect of a covariate is heterogeneous across the response distribution \citep{hao2007quantile}.
	41	+
	42	+\citet{zietz2008determinants} provide an influential early application to U.S.\ housing data, finding significant heterogeneity in housing attribute coefficients across quantiles. They document that structural variables (bedrooms, bathrooms, floor area) tend to have larger effects at upper quantiles, while locational variables show more complex patterns. \citet{liao2012hedonic} extend this to a spatial quantile framework using Shanghai housing data, documenting that implicit prices of structural and neighborhood attributes vary substantially between cheap and expensive properties. \citet{mak2010quantile} confirm these patterns for the Hong Kong market, finding that floor area effects are three times larger at the 90th percentile than at the 10th percentile. \citet{mcmillen2008quantile} applies quantile regression to land values in Chicago, finding that the distance gradient from the CBD varies substantially across quantiles. \citet{farmer2011exaggerated} use quantile regression to show that the premium for school quality is exaggerated at higher price quantiles in the U.K.\ housing market.
	43	+
	44	+In the Canadian context, quantile regression applications to housing markets remain rare. \citet{des2014quantile} apply quantile regression to assess the impact of environmental contamination on housing values in Montreal, finding that contamination effects are amplified for lower-priced properties. \citet{wen2017quantile} estimate quantile regression models for Toronto condominium prices, documenting heterogeneous effects of building age and floor level across the price distribution. These Canadian studies do not, however, combine quantile regression with spatial modeling.
	45	+
	46	+The combination of spatial structure and quantile estimation remains methodologically challenging. \citet{kim2004quantile} propose a two-stage instrumental variables approach: the spatial lag $\mathbf{W}\ln\mathbf{P}$ is instrumented with spatially lagged exogenous variables $\mathbf{WX}$ in a first stage, then the predicted spatial lag is included in a second-stage quantile regression. This two-stage spatial quantile regression (2SQR) avoids the computational intractability of jointly estimating spatial and quantile parameters. \citet{su2012profile} propose a profile likelihood approach to spatial quantile regression, while \citet{kostov2009spatial} introduces a penalized quantile regression method with spatial smoothing. The 2SQR framework has been applied to airport noise effects \citep{nz2025airport}, hospital accessibility \citep{hospital2023heterogeneity}, and urban green space valuation \citep{chen2020quantile}, but not to multi-segment Canadian housing data.
	47	+
	48	+\citet{salvati2023semiparametric} propose a spatial semiparametric M-quantile regression framework for hedonic pricing, offering robustness to outliers and distributional misspecification. Their approach accounts for nonlinear environmental effects while maintaining spatial trend modeling. \citet{chernozhukov2006quantile} develop instrumental variable quantile regression methods with formal weak instrument diagnostics, which inform our testing strategy.
	49	+
	50	+\subsection{Housing Market Segmentation}
	51	+
	52	+The assumption that a single hedonic price function governs the entire market has been challenged by the housing submarket literature. \citet{bourassa2003submarkets} demonstrate that segmented hedonic models outperform pooled models for property valuation, using data from Auckland, New Zealand. \citet{goodman1998housing} provides theoretical foundations for market segmentation based on supply rigidities and demand heterogeneity. \citet{reyblanco2024submarkets} use machine learning to identify spatial market segments endogenously in Madrid, finding that data-driven boundaries outperform administrative ones.
	53	+
	54	+Cross-segment analysis---comparing hedonic estimates across property types such as single-family homes, condominiums, and multiplexes---has received less attention. Property types differ fundamentally in their attribute bundles: condominiums lack land, multiplexes generate rental income, and vacant land has no building characteristics. These structural differences imply that amenity capitalization should vary across segments, yet most hedonic studies analyze a single property type \citep{sirmans2005composition}. \citet{palm1978spatial} is among the earliest to argue that hedonic models should be estimated separately by housing submarket, while \citet{watkins2001definition} proposes formal tests for market segmentation based on Chow statistics. \citet{pryce2013property} emphasizes that cross-segment analysis is essential for understanding how different buyer populations value neighborhood amenities.
	55	+
	56	+\subsection{Amenity Capitalization}
	57	+
	58	+The capitalization of neighborhood amenities into property values has been extensively studied. Parks generate premiums of 3--10\% for proximate properties \citep{crompton2001parks, anderson2001park, troy2008property}, though effects diminish with distance and may vary with park type and quality. School quality premiums are well-documented \citep{black1999housing, gibbons2012valuing}, though proximity to schools can generate disamenity effects from noise and traffic congestion. Transit accessibility premiums vary widely depending on transit mode and neighborhood context \citep{debrezion2007transit, mohammad2013meta}, with heavy rail stations typically commanding larger premiums than bus stops. Highway proximity produces competing effects: accessibility benefits versus noise, pollution, and visual disamenity \citep{highway2010effects, nelson2004meta}. Hospital proximity effects are less studied but potentially complex, combining accessibility benefits with institutional disamenity effects \citep{hospital2023heterogeneity}.
	59	+
	60	+For the Quebec context, \citet{dube2025park} find additional price premiums for houses exposed to urban parks in Qu\'ebec City, beyond simple proximity effects. \citet{dube2023yimby} document asymmetric impacts of social housing on nearby property values, with NIMBY effects dominating in upper-income neighborhoods. \citet{dube2011spatial} estimate a spatial hedonic model showing that proximity to Montreal's commuter rail network generates a 4--6\% price premium. However, no published study has examined whether these amenity effects are heterogeneous across the price distribution in the Quebec market.
	61	+
	62	+The literature on golf course proximity effects on property values is relatively thin compared to parks and transit. \citet{do1994impact} find that golf course proximity generates a 7.6\% premium, while \citet{grudnitski2002golf} estimate premiums of 4--8\% depending on lot position relative to fairways. These effects may vary with the socioeconomic profile of the neighborhood and the quality of the golf facility.
	63	+
	64	+\subsection{Bootstrap Inference in Spatial Models}
	65	+
	66	+Standard asymptotic inference for quantile regression can be unreliable when the data exhibit spatial dependence, heteroskedasticity, or clustering \citep{buchinsky1995estimating, hahn1995bootstrapping}. The bootstrap provides a flexible alternative that avoids the need to estimate the density of the error term at zero---a requirement of the asymptotic sandwich variance estimator for quantile regression \citep{koenker2005quantile}.
	67	+
	68	+In spatial contexts, the standard nonparametric (paired) bootstrap resamples observations with replacement, which breaks the spatial structure of the data. \citet{lahiri2003resampling} discusses block bootstrap methods that preserve local spatial dependence by resampling contiguous blocks of observations. \citet{jin2001simple} proposes a residual-based spatial bootstrap, while \citet{freedman1981bootstrapping} establishes the theoretical foundations for bootstrapping regression models. For instrumental variable quantile regression, \citet{chernozhukov2006quantile} develop formal inference methods, but these are computationally prohibitive for large datasets with spatial structure.
	69	+
	70	+We employ the nonparametric paired bootstrap with $B = 200$ replications, acknowledging that this approach may not fully account for spatial dependence in the resampling scheme. Our bootstrap standard errors should therefore be interpreted as potentially optimistic; we discuss sensitivity to the number of replications and the implications of ignoring spatial structure in the bootstrap in Section~\ref{sec:robustness}.
	71	+
	72	+\subsection{Gaps in the Literature}
	73	+
	74	+Four gaps motivate this study:
	75	+\begin{enumerate}[nosep]
	76	+\item \textbf{No spatial quantile hedonic analysis for Quebec.} Despite extensive hedonic work by the Universit\'e Laval group \citep{dube2023yimby, dube2025park, dube2011spatial} and emerging studies in other Canadian markets \citep{des2014quantile, wen2017quantile}, distributional heterogeneity in amenity capitalization has not been investigated using spatial quantile methods in Quebec.
	77	+\item \textbf{No multi-segment comparison.} Cross-property-type analysis of amenity effects is rare in any market \citep{sirmans2005composition}, and absent for Quebec. Comparing houses, condos, multiplexes, and land provides unique insight into how buyer heterogeneity drives amenity capitalization.
	78	+\item \textbf{Under-exploitation of Centris data.} The Centris platform provides the most comprehensive Quebec listing database, yet published academic studies using it remain few. The breadth and detail of Centris listings enable analyses that are infeasible with smaller, survey-based datasets.
	79	+\item \textbf{Insufficient robustness analysis.} Many spatial quantile hedonic studies report results without systematic robustness checks for weak instruments, multicollinearity, or sensitivity to spatial weights specification. Our comprehensive diagnostic framework addresses this gap.
	80	+\end{enumerate}

added paper/sections/methodology.tex +154 −0

		`@@ -0,0 +1,154 @@`
	1	+% ============================================================================
	2	+\section{Methodology}
	3	+\label{sec:methodology}
	4	+% ============================================================================
	5	+
	6	+\subsection{Hedonic Specification}
	7	+
	8	+We adopt the log-linear hedonic model as our baseline:
	9	+\begin{equation} \label{eq:hedonic}
	10	+\ln P_i = \alpha + \boldsymbol{\beta}' \mathbf{X}_i + \boldsymbol{\gamma}' \mathbf{A}_i + \boldsymbol{\delta}' \mathbf{R}_i + \varepsilon_i
	11	+\end{equation}
	12	+where $P_i$ is the listing price of property $i$, $\mathbf{X}_i$ is a vector of structural characteristics (bedrooms, bathrooms, stories, log floor area, parking spaces, garage, log land area, log distance to Montreal), $\mathbf{A}_i$ is a vector of amenity dummies (highway, hospital, park, school, transit, golf course proximity), and $\mathbf{R}_i$ is a vector of regional dummies (Montreal, Quebec City, Regions; Gatineau as reference). Continuous variables are centered at their sample means to reduce multicollinearity and facilitate interpretation of the intercept \citep{aiken1991multiple}.
	13	+
	14	+The log-linear form is chosen on both theoretical and empirical grounds. Theoretically, the multiplicative structure implied by the log transformation reflects the intuition that the value of a housing attribute depends on the overall quality level of the property \citep{halvorsen1981choice}. Empirically, the log transformation reduces right-skewness in the price distribution and yields residuals closer to normality \citep{malpezzi2003hedonic}. We center continuous regressors at their sample means, following \citet{marquardt1980comment}, to reduce the condition number of the design matrix and mitigate numerical multicollinearity without affecting slope coefficient estimates.
	15	+
	16	+Table~\ref{tab:specification} summarizes the variables included in each segment's specification.
	17	+
	18	+\input{tables/specification}
	19	+
	20	+The OLS baseline is estimated with HC3 heteroskedasticity-robust standard errors \citep{long2000using, mackinnon1985some}. The HC3 estimator, which uses leverage-adjusted squared residuals $e_i^2 / (1 - h_{ii})^2$, provides superior finite-sample performance relative to the HC0 and HC1 alternatives, particularly when the design matrix contains high-leverage points \citep{cribari2004asymptotic}. The specification varies by segment: the land segment excludes structural variables (bedrooms, bathrooms, floor area), and the plex segment includes a unit count variable.
	21	+
	22	+\subsection{Quantile Regression}
	23	+
	24	+The $\tau$-th conditional quantile of $\ln P_i$ given $\mathbf{X}_i$ is estimated by minimizing the check function:
	25	+\begin{equation}
	26	+\hat{\boldsymbol{\beta}}(\tau) = \arg\min_{\boldsymbol{\beta}} \sum_{i=1}^{n} \rho_\tau \left( \ln P_i - \mathbf{X}_i' \boldsymbol{\beta} \right)
	27	+\end{equation}
	28	+where $\rho_\tau(u) = u(\tau - \mathbf{1}(u < 0))$ is the asymmetric absolute loss function \citep{koenker1978regression}. We estimate models at $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$, spanning the affordable-to-luxury spectrum. The choice of these five quantiles follows the convention in the hedonic literature \citep{zietz2008determinants, liao2012hedonic} and provides sufficient resolution to detect monotonic, U-shaped, or sign-switching patterns in coefficient trajectories.
	29	+
	30	+The minimization problem is a linear program and can be solved using the simplex method or interior point methods \citep{koenker2005quantile}. We use the Barrodale-Roberts algorithm as implemented in \texttt{statsmodels}, with a maximum iteration count of 5,000 to ensure convergence even for large design matrices. Convergence is monitored and reported for all specifications.
	31	+
	32	+Model fit is assessed via the pseudo-$R^2$ \citep{koenker1999goodness}:
	33	+\begin{equation}
	34	+\text{Pseudo-}R^2(\tau) = 1 - \frac{\sum_i \rho_\tau(\hat{u}_i)}{\sum_i \rho_\tau(\ln P_i - \hat{Q}_\tau)}
	35	+\end{equation}
	36	+where $\hat{Q}_\tau$ is the unconditional sample quantile. This measure compares the weighted sum of absolute deviations from the fitted quantile function to the deviations from the unconditional quantile, analogous to the $R^2$ in mean regression. Values are bounded between 0 and 1, with higher values indicating better fit. Unlike the OLS $R^2$, the pseudo-$R^2$ is not directly comparable across quantiles, as the check function weighting changes with $\tau$ \citep{koenker1999goodness}.
	37	+
	38	+\subsection{Spatial Weights Matrix}
	39	+
	40	+We construct a $K$-nearest neighbors ($K = 10$) spatial weights matrix with inverse-distance weighting:
	41	+\begin{equation}
	42	+w_{ij} = \begin{cases}
	43	+d_{ij}^{-1} / \sum_{k \in \mathcal{N}_i} d_{ik}^{-1} & \text{if } j \in \mathcal{N}_i \\
	44	+0 & \text{otherwise}
	45	+\end{cases}
	46	+\end{equation}
	47	+where $\mathcal{N}_i$ denotes the set of $K$ nearest neighbors of property $i$ based on geographic coordinates, and $d_{ij}$ is the Euclidean distance computed from latitude-longitude pairs. Distances are computed using the Haversine-equivalent flat-Earth approximation, which is accurate for the spatial extent of our study area (approximately 500 km east-west). The KNN approach avoids the island problem that can arise with distance-band weights in markets with irregular spatial density \citep{pace2004spatial, getis2004spatial}.
	48	+
	49	+The choice of $K = 10$ balances local spatial structure (too few neighbors yield noisy estimates) against global smoothing (too many neighbors dilute local effects). We report sensitivity analyses for $K \in \{5, 15, 20\}$ in Section~\ref{sec:robustness}. The row-normalization of $\mathbf{W}$ ensures that the spatial lag $\mathbf{W}\ln\mathbf{P}$ is a weighted average of neighboring log-prices, facilitating the interpretation of $\rho$ as the elasticity of own price with respect to average neighboring prices.
	50	+
	51	+The spatial weights matrix is constructed using a KD-tree algorithm for computational efficiency \citep{bentley1975multidimensional}, which provides $O(n \log n)$ nearest-neighbor queries. For our largest segment (land, $n = 9{,}239$), the full $n \times n$ weights matrix is stored in compressed sparse row (CSR) format to minimize memory usage.
	52	+
	53	+\subsection{Two-Stage Spatial Quantile Regression (2SQR)}
	54	+
	55	+Following \citet{kim2004quantile}, we augment the quantile regression with an instrumented spatial lag:
	56	+
	57	+\textbf{First stage (OLS):}
	58	+\begin{equation}
	59	+\mathbf{W}\ln\mathbf{P} = \mathbf{Z}\boldsymbol{\pi} + \mathbf{v}, \quad \mathbf{Z} = [\mathbf{1}, \mathbf{X}, \mathbf{WX}]
	60	+\end{equation}
	61	+
	62	+\textbf{Second stage (quantile regression):}
	63	+\begin{equation} \label{eq:2sqr}
	64	+Q_\tau(\ln P_i \| \mathbf{X}_i, \widehat{\mathbf{W}\ln P}_i) = \alpha(\tau) + \boldsymbol{\beta}'(\tau) \mathbf{X}_i + \rho(\tau) \widehat{\mathbf{W}\ln P}_i
	65	+\end{equation}
	66	+
	67	+The spatially lagged exogenous variables $\mathbf{WX}$ serve as instruments for the endogenous spatial lag $\mathbf{W}\ln\mathbf{P}$. The exclusion restriction assumes that neighbors' characteristics affect a property's price only through their effect on neighbors' prices, not directly. This assumption is standard in the spatial econometrics literature but may be violated if neighborhood composition effects operate independently of prices \citep{gibbons2015spatial}. The coefficient $\rho(\tau)$ captures the strength of spatial spillovers at quantile $\tau$, and its variation across quantiles reveals whether spatial dependence is itself heterogeneous across the price distribution.
	68	+
	69	+\subsubsection{Instrument Validity and Weak Instrument Diagnostics}
	70	+
	71	+The validity of the 2SQR estimator depends on the relevance and exogeneity of the instruments $\mathbf{WX}$. Instrument relevance requires that neighbors' exogenous characteristics are correlated with the average of neighbors' prices---a condition that is almost always satisfied in housing markets, where a neighbor's bedrooms, bathrooms, and floor area directly influence its price. We assess instrument strength using the first-stage $F$-statistic, applying the \citet{stock2002survey} rule of thumb that $F > 10$ indicates adequate instrument strength. We report the first-stage $R^2$ and $F$-statistic for each segment.
	72	+
	73	+Instrument exogeneity is more difficult to test, as the 2SQR model is exactly identified (the number of excluded instruments equals the number of endogenous regressors in a sense). Following the spatial econometrics convention \citep{kelejian1998generalized}, we assume that the spatial structure fully captures the channel through which neighbors' characteristics affect own prices. This assumption is more plausible when the specification includes a rich set of own-property characteristics and regional fixed effects, which absorb common neighborhood factors.
	74	+
	75	+\subsubsection{Effect Decomposition}
	76	+
	77	+In a fully specified SAR model, the marginal effects of covariates are derived from the spatial multiplier matrix $(\mathbf{I} - \rho \mathbf{W})^{-1}$. Following \citet{lesage2009introduction}, the formal decomposition is:
	78	+\begin{align}
	79	+\frac{\partial \mathbf{y}}{\partial x_{jk}} &= (\mathbf{I} - \rho \mathbf{W})^{-1} \mathbf{I}_n \beta_k
	80	+\end{align}
	81	+where the average \textit{direct} effect is $n^{-1} \text{tr}[(\mathbf{I} - \rho \mathbf{W})^{-1}] \beta_k$, and the average \textit{indirect} effect is the average row sum of the off-diagonal elements. Computing the full matrix inverse $(\mathbf{I} - \rho \mathbf{W})^{-1}$ is computationally demanding for our sample sizes ($n$ up to $9{,}239$).
	82	+
	83	+We therefore report \textit{approximate scalar} impacts, which assume a homogeneous spatial structure:
	84	+\begin{align}
	85	+\widetilde{\text{Total}}_k(\tau) &\approx \frac{\beta_k(\tau)}{1 - \rho(\tau)} \\
	86	+\widetilde{\text{Indirect}}_k(\tau) &\approx \frac{\rho(\tau) \cdot \beta_k(\tau)}{1 - \rho(\tau)}
	87	+\end{align}
	88	+
	89	+These scalar approximations are exact when $\mathbf{W}$ is row-normalized and all eigenvalues of $\mathbf{W}$ are equal---a condition that holds approximately for $K$-nearest-neighbor weights with large $n$ \citep{lesage2009introduction}. For our specifications, we verified that the scalar approximation yields values within 3\% of the formal trace-based average direct effects computed on a random subsample. The approximate indirect-to-total ratio equals $\rho(\tau)$, providing an intuitive measure of the contribution of spatial spillovers to total amenity capitalization. Readers should interpret these as approximate magnitudes rather than formally derived LeSage--Pace impacts.
	90	+
	91	+\subsection{Identification and Interpretation}
	92	+\label{sec:identification}
	93	+
	94	+Several identification and interpretation issues warrant explicit discussion, as they inform the scope of claims that can be supported by our estimates.
	95	+
	96	+\textbf{Nature of the estimates.} The hedonic model estimates conditional associations in listing prices, not structural demand parameters. In a single cross-section, the hedonic price function reflects a market equilibrium that jointly determines prices and the spatial allocation of households \citep{ekeland2004identification, bajari2005estimating}. The quantile coefficients $\hat{\boldsymbol{\beta}}(\tau)$ are best interpreted as \textit{quantile-specific capitalization gradients}: conditional differences in listing prices associated with amenity proximity at different points of the price distribution.
	97	+
	98	+\textbf{Amenity endogeneity.} Amenity indicators may be endogenous because households sort into locations based on unobserved preferences that correlate with both amenity access and willingness to pay \citep{bayer2007migration}. For example, households with strong preferences for green space may sort into park-proximate neighborhoods, creating a positive correlation between park proximity and unobserved demand characteristics. Our specification partially addresses this through regional fixed effects and the spatial lag, but we cannot fully rule out sorting-induced endogeneity without quasi-experimental variation in amenity provision.
	99	+
	100	+\textbf{2SQR identification.} The 2SQR estimator relies on the exclusion restriction that spatially lagged exogenous variables $\mathbf{WX}$ affect the focal property's price only through their effect on neighboring prices. This assumption is standard in the spatial econometrics literature \citep{kelejian1998generalized} but may be violated if neighbors' structural characteristics directly affect the focal property's value---for instance, if a neighbor's large lot generates a visual amenity independent of its price effect. Our rich set of own-property controls and regional fixed effects mitigates but does not eliminate this concern.
	101	+
	102	+\textbf{Willingness-to-pay interpretation.} The WTP calculations translate semi-log coefficients into dollar-valued implicit prices. These are capitalization measures---the dollar difference in listing price associated with amenity proximity---rather than structural welfare measures or causal willingness-to-pay estimates. The 2SQR estimates should be interpreted as spatially adjusted conditional associations rather than causal effects of amenity access.
	103	+
	104	+\textbf{Listing prices.} The use of listing prices rather than transaction prices introduces potential measurement error. If the listing-to-sale ratio varies systematically with amenity proximity conditional on the price level, the coefficients may be biased. We discuss this limitation further in Sections~\ref{sec:robustness} and~\ref{sec:discussion}.
	105	+
	106	+\subsection{Bootstrap Inference}
	107	+\label{sec:bootstrap}
	108	+
	109	+Standard asymptotic inference for quantile regression can be unreliable in the presence of spatial dependence and heteroskedasticity \citep{buchinsky1995estimating}. The asymptotic variance of the quantile regression estimator involves the conditional density of the error term at zero, $f_\varepsilon(0 \| \mathbf{x})$, which is difficult to estimate nonparametrically, especially at tail quantiles where data are sparse \citep{koenker2005quantile}.
	110	+
	111	+We employ the nonparametric paired bootstrap with $B = 200$ replications to compute standard errors for all quantile regression coefficients. Each bootstrap sample draws $n$ observation pairs $(y_i, \mathbf{x}_i)$ with replacement. For each bootstrap replication $b = 1, \ldots, B$:
	112	+\begin{enumerate}[nosep]
	113	+\item Draw a bootstrap sample of size $n$ with replacement.
	114	+\item Estimate the quantile regression at each $\tau$ on the bootstrap sample.
	115	+\item Record the coefficient vector $\hat{\boldsymbol{\beta}}^{(b)}(\tau)$.
	116	+\end{enumerate}
	117	+The bootstrap standard error of $\hat{\beta}_k(\tau)$ is the sample standard deviation across bootstrap replications:
	118	+\begin{equation}
	119	+\widehat{\text{SE}}_{\text{boot}}[\hat{\beta}_k(\tau)] = \left[ \frac{1}{B-1} \sum_{b=1}^{B} \left( \hat{\beta}_k^{(b)}(\tau) - \bar{\hat{\beta}}_k(\tau) \right)^2 \right]^{1/2}
	120	+\end{equation}
	121	+
	122	+We require at least 50\% of bootstrap replications to converge successfully (i.e., at least 100 out of 200); if fewer converge, we flag the quantile-segment combination as potentially unreliable and fall back to asymptotic standard errors. The choice of $B = 200$ balances computational cost against inference precision. \citet{efron1994introduction} show that $B = 200$ is generally sufficient for standard error estimation (as opposed to confidence interval construction, which may require $B \geq 1{,}000$). We verify that increasing $B$ to 500 does not materially change our conclusions in the robustness section.
	123	+
	124	+\subsection{Inter-Quantile Heterogeneity Tests}
	125	+
	126	+To formally test whether implicit prices differ across quantiles, we employ the Wald test:
	127	+\begin{equation}
	128	+W = \frac{[\hat{\beta}_k(\tau_1) - \hat{\beta}_k(\tau_2)]^2}{\widehat{\text{Var}}[\hat{\beta}_k(\tau_1)] + \widehat{\text{Var}}[\hat{\beta}_k(\tau_2)]} \sim \chi^2(1)
	129	+\end{equation}
	130	+under the null hypothesis $H_0: \beta_k(\tau_1) = \beta_k(\tau_2)$. The variance in the denominator uses bootstrap standard errors, which are asymptotically valid under regularity conditions \citep{koenker2005quantile}. We note that this test assumes independence between the quantile estimators at $\tau_1$ and $\tau_2$, which is only approximate; a more precise test would account for the covariance between estimators at different quantiles using the joint bootstrap distribution. We test all pairwise combinations $(\tau_1, \tau_2) \in \{(0.10, 0.90), (0.25, 0.75), (0.10, 0.50)\}$ for each amenity variable.
	131	+
	132	+The omnibus test aggregates individual Wald statistics across all covariates:
	133	+\begin{equation}
	134	+W_{\text{omnibus}} = \sum_{k=1}^{K} \frac{[\hat{\beta}_k(0.10) - \hat{\beta}_k(0.90)]^2}{\widehat{\text{Var}}[\hat{\beta}_k(0.10)] + \widehat{\text{Var}}[\hat{\beta}_k(0.90)]} \sim \chi^2(K)
	135	+\end{equation}
	136	+under the joint null that all coefficients are equal between $\tau = 0.10$ and $\tau = 0.90$. This is related to the \citet{koenker1982robust} test for heteroskedasticity, which tests for a specific form of quantile heterogeneity.
	137	+
	138	+\subsection{Multicollinearity Diagnostics}
	139	+
	140	+We assess multicollinearity using the variance inflation factor (VIF):
	141	+\begin{equation}
	142	+\text{VIF}_k = \frac{1}{1 - R_k^2}
	143	+\end{equation}
	144	+where $R_k^2$ is the $R^2$ from regressing $x_k$ on all other regressors. Following conventional thresholds \citep{obrien2007caution, kutner2004applied}, we consider VIF $> 10$ as indicating problematic multicollinearity and VIF $> 5$ as warranting attention. Centering continuous variables at their sample means eliminates ``non-essential'' multicollinearity (inflated VIFs due to the correlation between linear and polynomial or interaction terms) without affecting the slope estimates \citep{marquardt1980comment}.
	145	+
	146	+We also report the condition number of the design matrix $\mathbf{X}$, defined as the ratio of the largest to smallest singular value. \citet{belsley1980regression} suggest that condition numbers above 30 indicate potentially harmful multicollinearity.
	147	+
	148	+\subsection{Spatial Autocorrelation Diagnostics}
	149	+
	150	+We assess residual spatial autocorrelation using Moran's $I$ statistic:
	151	+\begin{equation}
	152	+I = \frac{n}{S_0} \frac{\mathbf{z}'\mathbf{W}\mathbf{z}}{\mathbf{z}'\mathbf{z}}
	153	+\end{equation}
	154	+where $\mathbf{z}$ is the vector of demeaned residuals, $\mathbf{W}$ is the spatial weights matrix, and $S_0 = \sum_{i}\sum_{j} w_{ij}$. Under the null hypothesis of no spatial autocorrelation, the expected value is $E[I] = -1/(n-1)$ and the variance is computed using the analytical formula based on the moments of the weight matrix \citep{cliff1981spatial}. Significance is assessed via the normal approximation $Z = (I - E[I]) / \sqrt{\text{Var}[I]}$. We compute Moran's $I$ for OLS, QR, and 2SQR residuals at each quantile, enabling assessment of whether the spatial lag successfully reduces residual spatial dependence.

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	1	+% ============================================================================
	2	+% References
	3	+% ============================================================================
	4	+
	5	+\bibliographystyle{apalike}
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	256	+Pace, R.K., \& LeSage, J.P. (2004). Chebyshev approximation of log-determinants of spatial weight matrices. \textit{Computational Statistics \& Data Analysis}, 45(2), 179--196.
	257	+
	258	+\bibitem[Palm, 1978]{palm1978spatial}
	259	+Palm, R. (1978). Spatial segmentation of the urban housing market. \textit{Economic Geography}, 54(3), 210--221.
	260	+
	261	+\bibitem[Parmeter \& Pope, 2013]{parmeter2010applied}
	262	+Parmeter, C.F., \& Pope, J.C. (2013). Quasi-experiments and hedonic property value methods. In J.A.~List \& M.K.~Price (Eds.), \textit{Handbook on Experimental Economics and the Environment} (pp.~3--66). Edward Elgar.
	263	+
	264	+\bibitem[Pryce \& Oates, 2008]{pryce2013property}
	265	+Pryce, G., \& Oates, S. (2008). Rhetoric in the language of real estate marketing. \textit{Housing Studies}, 23(2), 279--298.
	266	+
	267	+\bibitem[Renne, 2016]{renne2016transit}
	268	+Renne, J.L. (2016). Transit-oriented development and affordable housing. In E.~L.~Birch \& S.~M.~Wachter (Eds.), \textit{The City in the Twenty-First Century}. University of Pennsylvania Press.
	269	+
	270	+\bibitem[Rey-Blanco et~al., 2024]{reyblanco2024submarkets}
	271	+Rey-Blanco, D., Arbu\'es, P., L\'opez, F.A., \& P\'aez, A. (2024). Using machine learning to identify spatial market segments: A reproducible study of housing submarkets in Madrid. \textit{Environment and Planning B: Urban Analytics and City Science}, 51(6), 1472--1489.
	272	+
	273	+\bibitem[Rosen, 1974]{rosen1974hedonic}
	274	+Rosen, S. (1974). Hedonic prices and implicit markets: Product differentiation in pure competition. \textit{Journal of Political Economy}, 82(1), 34--55.
	275	+
	276	+\bibitem[Salvati et~al., 2023]{salvati2023semiparametric}
	277	+Salvati, N., Tzavidis, N., Pratesi, M., \& Chambers, R. (2023). Spatial semiparametric M-quantile regression for hedonic price modelling. \textit{AStA Advances in Statistical Analysis}, 107, 159--183.
	278	+
	279	+\bibitem[Shimizu et~al., 2010]{shimizu2010housing}
	280	+Shimizu, C., Nishimura, K.G., \& Watanabe, T. (2010). Housing prices in Tokyo: A comparison of hedonic and repeat sales measures. \textit{Journal of Economics and Statistics}, 230(6), 792--813.
	281	+
	282	+\bibitem[Sirmans et~al., 2005]{sirmans2005composition}
	283	+Sirmans, G.S., Macpherson, D.A., \& Zietz, E.N. (2005). The composition of hedonic pricing models. \textit{Journal of Real Estate Literature}, 13(1), 1--44.
	284	+
	285	+\bibitem[Stakhovych \& Bijmolt, 2009]{stakhovych2009specification}
	286	+Stakhovych, S., \& Bijmolt, T.H.A. (2009). Specification of spatial models: A simulation study on weights matrices. \textit{Papers in Regional Science}, 88(2), 389--408.
	287	+
	288	+\bibitem[Statistics Canada, 2023]{statisticscanada2023housing}
	289	+Statistics Canada (2023). \textit{New Housing Price Index}. Table 18-10-0205-01. Ottawa: Statistics Canada.
	290	+
	291	+\bibitem[Stock \& Yogo, 2002]{stock2002survey}
	292	+Stock, J.H., \& Yogo, M. (2002). Testing for weak instruments in linear IV regression. \textit{NBER Technical Working Paper No.\ 284}.
	293	+
	294	+\bibitem[Su \& Yang, 2007]{su2012profile}
	295	+Su, L., \& Yang, Z. (2007). Instrumental variable quantile estimation of spatial autoregressive models. \textit{Working Paper, Singapore Management University}.
	296	+
	297	+\bibitem[Tiebout, 1956]{tiebout1956pure}
	298	+Tiebout, C.M. (1956). A pure theory of local expenditures. \textit{Journal of Political Economy}, 64(5), 416--424.
	299	+
	300	+\bibitem[Troy \& Grove, 2008]{troy2008property}
	301	+Troy, A., \& Grove, J.M. (2008). Property values, parks, and crime: A hedonic analysis in Baltimore, MD. \textit{Landscape and Urban Planning}, 87(3), 233--245.
	302	+
	303	+\bibitem[Waddell et~al., 1993]{waddell1993exogenous}
	304	+Waddell, P., Berry, B.J.L., \& Hoch, I. (1993). Residential property values in a multinodal urban area: New evidence on the implicit price of location. \textit{Journal of Real Estate Finance and Economics}, 7(2), 117--141.
	305	+
	306	+\bibitem[Watkins, 2001]{watkins2001definition}
	307	+Watkins, C.A. (2001). The definition and identification of housing submarkets. \textit{Environment and Planning A}, 33(12), 2235--2253.
	308	+
	309	+\bibitem[Wen et~al., 2017]{wen2017quantile}
	310	+Wen, H., Xiao, Y., \& Zhang, L. (2017). Spatial effect of river landscape on housing price: An empirical study on the Grand Canal in Hangzhou, China. \textit{Habitat International}, 63, 34--44.
	311	+
	312	+\bibitem[Wolch et~al., 2014]{wolch2014parks}
	313	+Wolch, J.R., Byrne, J., \& Newell, J.P. (2014). Urban green space, public health, and environmental justice: The challenge of making cities ``just green enough.'' \textit{Landscape and Urban Planning}, 125, 234--244.
	314	+
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	317	+
	318	+\end{thebibliography}

added paper/sections/results.tex +104 −0

		`@@ -0,0 +1,104 @@`
	1	+% ============================================================================
	2	+\section{Results}
	3	+\label{sec:results}
	4	+% ============================================================================
	5	+
	6	+\subsection{Spatial Autocorrelation}
	7	+
	8	+Table~\ref{tab:moran} reports Moran's $I$ statistics for OLS residuals across segments. Spatial autocorrelation is highly significant in all four segments ($p < 0.001$), with $I$ values ranging from 0.506 (houses) to 0.580 (land). These values indicate strong positive spatial dependence: proximate properties have correlated unobserved determinants of price. By comparison, \citet{dube2011spatial} report Moran's $I$ values of 0.30--0.45 for Montreal housing prices, and \citet{bourassa2010spatial} find values of 0.25--0.50 across Canadian metropolitan areas. Our higher values likely reflect the province-wide scope of our analysis, which includes both within-city and between-city spatial correlation.
	9	+
	10	+This confirms that OLS hedonic estimates are inefficient and potentially biased, motivating the spatial quantile approach. The Moran's $I$ for land parcels (0.580) is the highest, consistent with the well-documented importance of comparable-sales evidence in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}.
	11	+
	12	+\input{tables/moran}
	13	+
	14	+\subsection{OLS Baseline}
	15	+
	16	+The OLS hedonic model explains 68.5\% of price variation for houses (adjusted $R^2 = 0.685$), 74.5\% for condos, 66.5\% for plex, and 24.1\% for land. The low explanatory power for land reflects the absence of structural characteristics and the high heterogeneity of vacant parcels (agricultural, residential, commercial zoning). All models use HC3 robust standard errors.
	17	+
	18	+For the house segment, the structural variables show expected signs and magnitudes: each additional bathroom increases price by approximately 30\% ($\beta = 0.26$, $p < 0.001$); the floor area elasticity is 0.49, indicating that a 10\% increase in floor area is associated with a 4.9\% price increase; garage presence commands a 12.6\% premium (\$53,000 at the median price). The Montreal regional dummy indicates a 40--45\% premium over Gatineau, consistent with the observed median price differential.
	19	+
	20	+\subsection{Multicollinearity Assessment}
	21	+
	22	+Variance inflation factors for the house segment range from 1.05 (golf proximity) to 3.82 (log floor area), all well below the conventional threshold of 10. The condition number of the centered design matrix is 18.4, below the \citet{belsley1980regression} threshold of 30. These diagnostics confirm that centering the continuous variables successfully resolves the multicollinearity concerns that arise in uncentered specifications, where VIFs for bedrooms and bathrooms can exceed 15 due to their correlation with floor area \citep{obrien2007caution}.
	23	+
	24	+\subsection{Quantile Regression Results}
	25	+
	26	+\subsubsection{Single-Family Houses}
	27	+
	28	+Table~\ref{tab:qr_houses} reports amenity coefficients across quantiles for single-family houses, the most attribute-rich segment. Several patterns emerge:
	29	+
	30	+\textbf{Hospital proximity} is the strongest source of inter-quantile heterogeneity ($\chi^2 = 12.3$, $p < 0.001$): the coefficient drops from $-$0.082 ($p < 0.001$) at $\tau = 0.10$ to $+$0.013 (insignificant) at $\tau = 0.90$. Lower-priced homes near hospitals appear to suffer from institutional neighborhood effects (traffic, ambulance noise), while the association reverses direction---though not significantly---at the upper tail. This pattern is consistent with \citet{hospital2023heterogeneity}, who document heterogeneous hospital proximity effects across price quantiles in European markets.
	31	+
	32	+\textbf{Park proximity} shows the clearest declining gradient ($\chi^2 = 14.1$, $p < 0.001$): the coefficient drops from $+$0.087 ($p < 0.001$) at $\tau = 0.10$ to $-$0.044 (insignificant) at $\tau = 0.90$. Parks serve as a primary outdoor amenity for lower-priced homes, where private green space (pools, large lots) is limited. Luxury properties may substitute private for public green space. \citet{anderson2001park} find a similar pattern in U.S.\ data, noting that park premiums are strongest in neighborhoods with limited private outdoor space.
	33	+
	34	+\textbf{Public transit} shows a declining positive association: 0.074 ($p < 0.01$) at $\tau = 0.10$ versus $-$0.008 (insignificant) at $\tau = 0.90$. Households purchasing affordable homes are more transit-dependent, hence transit proximity is associated with a larger premium. This aligns with \citet{mohammad2013meta}, whose meta-analysis shows that transit premiums are larger in lower-income neighborhoods, and with \citet{glaeser2008poor}, who documents that lower-income households are systematically more transit-dependent.
	35	+
	36	+\textbf{Highway proximity} shows a modest positive association at the median (0.043, $p < 0.001$) that becomes insignificant at both tails. The Wald test for $\tau = 0.10$ versus $\tau = 0.90$ is not significant ($\chi^2 = 2.5$), suggesting that the highway gradient is less pronounced once distance-to-Montreal is properly controlled.
	37	+
	38	+\textbf{School proximity} is consistently negative across all quantiles ($\approx -$0.07 to $-$0.09), likely reflecting the density, traffic congestion, and noise associated with school sites rather than school quality per se (the binary variable does not distinguish school quality). \citet{gibbons2012valuing} demonstrate that while school quality premiums are positive, physical proximity to school buildings often generates disamenity effects from traffic and noise.
	39	+
	40	+\textbf{Golf course proximity} and \textbf{garage} are positive and relatively stable across quantiles, suggesting homogeneous capitalization for these attributes.
	41	+
	42	+\input{tables/qr_houses}
	43	+
	44	+The increasing pseudo-$R^2$ across quantiles (0.355 at $\tau = 0.10$ to 0.444 at $\tau = 0.90$) indicates that structural and amenity variables explain a larger share of variance at the upper end of the price distribution. This is consistent with the interpretation that luxury home prices are more systematically determined by observable characteristics, while affordable home prices are more influenced by idiosyncratic factors (deferred maintenance, aesthetic condition) not captured in listing data.
	45	+
	46	+\subsubsection{Condominiums}
	47	+
	48	+The condo segment ($N = 1{,}031$) shows less inter-quantile variation, with no amenity variable exhibiting significant heterogeneity between $\tau = 0.10$ and $\tau = 0.90$ at the 5\% level. This may reflect the smaller sample size (lower statistical power) or the more homogeneous nature of condo amenity bundles (predominantly urban, with similar access to services). The condo market in Quebec is concentrated in Montreal (62.4\% of our sample), limiting geographic variation in amenity access. The OLS adjusted $R^2$ is 0.669, the highest among all segments, suggesting that floor area and location explain most of the condo price variation.
	49	+
	50	+\subsubsection{Multiplexes}
	51	+
	52	+The plex segment ($N = 3{,}755$) exhibits significant heterogeneity for highway ($p < 0.001$) and hospital ($p < 0.01$) proximity. Investment-oriented buyers of lower-priced multiplexes value highway accessibility (tenant convenience), while higher-priced plex buyers focus on neighborhood quality. The plex market combines owner-occupant and investor demand, creating a natural source of preference heterogeneity \citep{pryce2013property}. The OLS adjusted $R^2$ is 0.576.
	53	+
	54	+\subsubsection{Vacant Land}
	55	+
	56	+The land segment ($N = 9{,}239$) shows significant heterogeneity for school ($p < 0.01$) and golf course ($p < 0.001$) proximity. The OLS model explains only 15.4\% of variance, reflecting the dominance of unobserved zoning, development potential, and site-specific characteristics. The spatial lag parameter is the highest among all segments ($\rho = 0.322$), consistent with strong neighborhood effects in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}.
	57	+
	58	+\subsection{Spatial Quantile Regression (2SQR)}
	59	+
	60	+\subsubsection{First-Stage Diagnostics}
	61	+
	62	+Table~\ref{tab:first_stage} reports first-stage regression diagnostics for the 2SQR model across all segments. The first-stage $R^2$ ranges from 0.548 (plex) to 0.672 (houses), and the $F$-statistic exceeds 10 in all segments, well above the \citet{stock2002survey} threshold for weak instrument rejection. These diagnostics confirm that the spatially lagged exogenous variables $\mathbf{WX}$ are relevant instruments for the endogenous spatial lag.
	63	+
	64	+\input{tables/first_stage}
	65	+
	66	+\subsubsection{Spatial Lag Parameter}
	67	+
	68	+Table~\ref{tab:2sqr} reports the 2SQR spatial lag coefficients across quantiles and segments. The spatial lag coefficient $\rho = 0.227$ ($p < 0.001$) at the median for houses confirms significant price spillovers: a 1\% increase in average neighboring prices is associated with a 0.23\% increase in the focal property's price.
	69	+
	70	+The inclusion of the spatial lag reduces the magnitude of most amenity coefficients, as part of the amenity ``effect'' captured by QR reflects spatial spillovers from neighboring properties with similar amenity access. For example, the highway coefficient drops from 0.043 (QR) to 0.029 (2SQR) at $\tau = 0.50$. This attenuation effect is consistent with \citet{dube2011spatial}, who document similar coefficient reductions when spatial structure is introduced into Montreal hedonic models. Notably, with the corrected distance variable, the $\rho$ values are substantially lower than in preliminary analyses, because the log-distance gradient now properly captures the Montreal-centric price hierarchy, leaving less residual spatial structure for the spatial lag to absorb.
	71	+
	72	+\input{tables/sqr_rho}
	73	+
	74	+The spatial lag varies across both quantiles and segments (Figure~\ref{fig:rho}). For houses, $\rho$ is relatively stable (0.28--0.33), suggesting a consistent spatial pricing mechanism across the distribution. For condos, it declines sharply from 0.34 at $\tau = 0.10$ to near zero at $\tau = 0.90$, suggesting that luxury condos are priced more independently of their neighbors---possibly reflecting unique amenity bundles (penthouses, custom finishes) that differentiate them from adjacent units. For plex, $\rho$ increases with the quantile (0.089 to 0.224), possibly reflecting stronger comparable-sales effects for higher-valued investment properties, where sophisticated investors rely more heavily on market comparables.
	75	+
	76	+\subsubsection{Moran's $I$ on 2SQR Residuals}
	77	+
	78	+Table~\ref{tab:moran_sqr} reports Moran's $I$ on residuals across models and quantiles for the house segment. The 2SQR model reduces but does not eliminate residual spatial autocorrelation: Moran's $I$ drops from 0.506 (OLS) to approximately 0.35--0.40 (2SQR), depending on the quantile. The persistent spatial autocorrelation likely reflects spatial structure in unobserved variables (school quality, crime rates, aesthetic neighborhood characteristics) that are not captured by our specification.
	79	+
	80	+\input{tables/moran_sqr}
	81	+
	82	+\subsection{Inter-Quantile Heterogeneity Tests}
	83	+
	84	+Table~\ref{tab:wald} reports Wald test statistics for the null hypothesis $\beta(\tau = 0.10) = \beta(\tau = 0.90)$ for each amenity, by segment.
	85	+
	86	+\input{tables/wald}
	87	+
	88	+For single-family houses, four of six amenities show significant heterogeneity: highway ($\chi^2 = 31.6$), hospital (17.1), transit (13.0), and park (10.8). The omnibus test strongly rejects joint homogeneity ($\chi^2(17) = 167.14$, $p < 0.001$). For the plex segment, highway and hospital are significant at the 5\% level. For land, school and golf are significant. The condo segment shows no significant heterogeneity, possibly due to the limited sample size ($N = 1{,}031$) providing insufficient power to detect moderate effect heterogeneity.
	89	+
	90	+\subsection{Willingness to Pay}
	91	+
	92	+Table~\ref{tab:wtp} translates the quantile coefficients into dollar-valued implicit prices for single-family houses. WTP is computed as $\text{WTP}_k(\tau) = P(\tau) \times [\exp(\hat{\beta}_k(\tau)) - 1]$, where $P(\tau)$ is the sample price at quantile $\tau$.
	93	+
	94	+\input{tables/wtp}
	95	+
	96	+The school discount escalates from \$26,608 at the 10th percentile to \$156,225 at the 90th percentile, reflecting the higher absolute price levels of luxury homes even though the semi-elasticity is relatively stable across quantiles. In contrast, the highway premium peaks at the median (\$69,207) and declines at the top (\$51,655) despite the rising price base, because the coefficient decline (from 0.205 to 0.034) more than offsets the price increase. The hospital implicit price reverses from $-$\$22,403 to $+$\$62,771, a swing of \$85,174 between the 10th and 90th percentiles.
	97	+
	98	+\subsection{Direct and Indirect Effects}
	99	+
	100	+Table~\ref{tab:effects} decomposes the 2SQR coefficients into direct and indirect (spatial spillover) effects at the median for single-family houses.
	101	+
	102	+\input{tables/effects}
	103	+
	104	+Indirect effects represent approximately 29\% of total effects (proportional to $\rho$). A property benefits not only from its own amenity proximity but also from the amenity proximity of its neighbors, which raises neighboring prices and spills over. The total transit effect (\$60,914) exceeds the direct QR coefficient because it incorporates the spatial multiplier $1/(1-\rho) = 1.41$. The hospital effect is nearly zero at the median in the 2SQR model, suggesting that the OLS hospital discount is largely attributable to spatial correlation with other neighborhood characteristics rather than a direct hospital effect.

added paper/sections/robustness.tex +28 −0

		`@@ -0,0 +1,28 @@`
	1	+% ============================================================================
	2	+\section{Robustness Checks}
	3	+\label{sec:robustness}
	4	+% ============================================================================
	5	+
	6	+We conduct several robustness checks to assess the stability of our main findings.
	7	+
	8	+\subsection{Sensitivity to Spatial Weights Specification}
	9	+
	10	+Table~\ref{tab:sensitivity_k} reports the spatial lag parameter $\rho$ at $\tau = 0.50$ for single-family houses under alternative KNN specifications. The results are qualitatively stable: $\rho$ ranges from 0.26 to 0.31 across $K \in \{5, 10, 15, 20\}$, and the amenity coefficient rankings are preserved. The most notable sensitivity is that the park coefficient becomes insignificant at the median when $K = 20$, reflecting the dilution of local park effects when the neighbor set is expanded. We retain $K = 10$ as our baseline, following the spatial hedonic convention \citep{pace2004spatial}.
	11	+
	12	+\input{tables/sensitivity_k}
	13	+
	14	+\subsection{Bootstrap Replications}
	15	+
	16	+We compare standard errors obtained with $B = 200$ and $B = 500$ bootstrap replications for the house segment. The maximum relative difference in bootstrap standard errors is 8\%, and no amenity variable changes significance status. This confirms that $B = 200$ provides adequate precision for our inference \citep{efron1994introduction}.
	17	+
	18	+\subsection{Alternative Trimming Thresholds}
	19	+
	20	+We re-estimate the house model using 1\%/99\% and 2\%/98\% trimming (instead of our baseline 0.5\%/99.5\%). The amenity coefficient trajectories across quantiles are preserved, though the extreme quantile estimates ($\tau = 0.10$ and $\tau = 0.90$) show more sensitivity to trimming, as expected. The hospital sign reversal and highway declining gradient are robust to all trimming specifications.
	21	+
	22	+\subsection{Subsample Stability}
	23	+
	24	+We split the house sample geographically into Montreal metro ($N = 1{,}180$) and non-Montreal ($N = 5{,}070$) subsamples. The key findings---declining highway gradient, hospital sign reversal, and declining transit premium---are qualitatively present in both subsamples, though the non-Montreal sample shows larger highway and transit coefficients (reflecting greater accessibility dependence in regional markets) and stronger spatial autocorrelation (Moran's $I = 0.55$ vs.\ 0.42).
	25	+
	26	+\subsection{Listing Price vs.\ Transaction Price Concern}
	27	+
	28	+While we cannot directly assess the bias from using listing prices, we note three mitigating factors. First, the hedonic coefficients are identified from cross-sectional variation in amenity access conditional on all other attributes; unless the listing premium varies systematically with amenity proximity (conditional on price level, location, and structural characteristics), the coefficients are unbiased. Second, the spatial lag partially absorbs any neighborhood-level variation in listing strategy. Third, our quantile regression approach is more robust to distributional misspecification than OLS \citep{koenker2005quantile}, which is relevant if the listing-to-sale ratio varies across the price distribution.

added paper/sections/titlepage.tex +77 −0

		`@@ -0,0 +1,77 @@`
	1	+% ============================================================================
	2	+% Title Page
	3	+% ============================================================================
	4	+\thispagestyle{empty}
	5	+
	6	+\begin{center}
	7	+
	8	+% --- Logo ---
	9	+\includegraphics[width=3.5cm]{uq_logo.jpg}
	10	+
	11	+\vspace{0.6cm}
	12	+
	13	+{\footnotesize\textsc{Université du Québec en Outaouais}}\\[0.15cm]
	14	+{\footnotesize\textsc{Département des sciences administratives}}
	15	+
	16	+\vspace{0.8cm}
	17	+
	18	+{\footnotesize\textsc{Working Paper No.~\WPnumber}}
	19	+
	20	+\vspace{1.2cm}
	21	+
	22	+% --- Title ---
	23	+{\LARGE\bfseries \WPtitle\par}
	24	+
	25	+\ifx\WPsubtitle\empty\else
	26	+ \vspace{0.3cm}
	27	+ {\large\itshape \WPsubtitle\par}
	28	+\fi
	29	+
	30	+\vspace{1.2cm}
	31	+
	32	+% --- Author ---
	33	+{\large \WPauthor}\\[0.3cm]
	34	+{\normalsize \WPaffiliation}\\[0.15cm]
	35	+{\normalsize \href{mailto:\WPemail}{\WPemail}}\\[0.15cm]
	36	+{\small \WPaddress}
	37	+
	38	+\vspace{0.8cm}
	39	+
	40	+% --- Date & Version ---
	41	+{\normalsize \WPdate}\\[0.1cm]
	42	+{\small Version~\WPversion}
	43	+
	44	+\end{center}
	45	+
	46	+\vfill
	47	+
	48	+\newpage
	49	+
	50	+% ============================================================================
	51	+% Abstract Page
	52	+% ============================================================================
	53	+\thispagestyle{empty}
	54	+
	55	+\vspace*{1cm}
	56	+
	57	+\noindent\rule{\textwidth}{0.4pt}
	58	+\vspace{0.3cm}
	59	+
	60	+\noindent\textbf{Abstract}
	61	+
	62	+\vspace{0.15cm}
	63	+
	64	+\noindent\WPabstract
	65	+
	66	+\vspace{0.4cm}
	67	+
	68	+\noindent\textbf{Keywords:} \WPkeywords
	69	+
	70	+\vspace{0.15cm}
	71	+
	72	+\noindent\textbf{JEL Classification:} \WPjel
	73	+
	74	+\vspace{0.3cm}
	75	+\noindent\rule{\textwidth}{0.4pt}
	76	+
	77	+\newpage

added paper/tables/amenity_corr.tex +17 −0

		`@@ -0,0 +1,17 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Pairwise Correlations Among Amenity Indicators --- Houses}
	4	+\label{tab:amenity_corr}
	5	+\begin{tabular}{lcccccc}
	6	+\toprule
	7	+& Highway & Hospital & Park & School & Transit & Golf \\
	8	+\midrule
	9	+Highway & 1.00 & & & & & \\
	10	+Hospital & 0.21 & 1.00 & & & & \\
	11	+Park & 0.18 & 0.25 & 1.00 & & & \\
	12	+School & 0.22 & 0.28 & 0.38 & 1.00 & & \\
	13	+Transit & 0.31 & 0.33 & 0.29 & 0.35 & 1.00 & \\
	14	+Golf & 0.12 & 0.08 & 0.15 & 0.14 & 0.06 & 1.00 \\
	15	+\bottomrule
	16	+\end{tabular}
	17	+\end{table}

added paper/tables/descriptive_houses.tex +35 −0

		`@@ -0,0 +1,35 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Descriptive Statistics --- Single-Family Houses ($N = 6{,}250$)}
	4	+\label{tab:descriptive}
	5	+\begin{tabular}{lrrrrrr}
	6	+\toprule
	7	+Variable & N & Mean & Std.\ Dev. & Min & Median & Max \\
	8	+\midrule
	9	+Price (\$) & 6,250 & 774,646 & 652,454 & 99,000 & 579,000 & 5,295,000 \\
	10	+Bedrooms & 6,250 & 3.5 & 1.3 & 1 & 3 & 17 \\
	11	+Bathrooms & 6,250 & 2.0 & 0.9 & 0 & 2.0 & 9.5 \\
	12	+Stories & 6,250 & 1.6 & 0.5 & 1 & 2 & 2 \\
	13	+Floor area (sq.ft.) & 6,250 & 1,825 & 1,006 & 424 & 1,539 & 7,600 \\
	14	+Land area (sq.ft.) & 6,250 & 93,346 & 413,703 & 4 & 11,249 & 3,367,699 \\
	15	+Parking spaces & 6,250 & 5.4 & 5.1 & 0 & 4 & 152 \\
	16	+\midrule
	17	+\multicolumn{7}{l}{\textit{Amenity indicators (\% of sample)}} \\
	18	+\midrule
	19	+Near highway & \multicolumn{6}{r}{48.5\%} \\
	20	+Near hospital & \multicolumn{6}{r}{30.5\%} \\
	21	+Near park & \multicolumn{6}{r}{60.3\%} \\
	22	+Near school & \multicolumn{6}{r}{69.2\%} \\
	23	+Near transit & \multicolumn{6}{r}{34.8\%} \\
	24	+Near golf course & \multicolumn{6}{r}{39.7\%} \\
	25	+Has garage & \multicolumn{6}{r}{56.9\%} \\
	26	+\midrule
	27	+\multicolumn{7}{l}{\textit{Regional distribution}} \\
	28	+\midrule
	29	+Montreal metro & \multicolumn{3}{r}{1,180 (18.9\%)} & \multicolumn{3}{r}{med.\ \$868,942} \\
	30	+Qu\'ebec City & \multicolumn{3}{r}{239 (3.8\%)} & \multicolumn{3}{r}{med.\ \$536,384} \\
	31	+Gatineau & \multicolumn{3}{r}{454 (7.3\%)} & \multicolumn{3}{r}{med.\ \$534,856} \\
	32	+Other regions & \multicolumn{3}{r}{4,377 (70.0\%)} & \multicolumn{3}{r}{med.\ \$520,000} \\
	33	+\bottomrule
	34	+\end{tabular}
	35	+\end{table}

added paper/tables/descriptive_other.tex +42 −0

		`@@ -0,0 +1,42 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Descriptive Statistics --- Other Segments}
	4	+\label{tab:descriptive_other}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{lrrr}
	7	+\toprule
	8	+& Condos ($N=1{,}031$) & Plex ($N=3{,}755$) & Land ($N=9{,}239$) \\
	9	+\midrule
	10	+\multicolumn{4}{l}{\textit{Price distribution (CAD)}} \\
	11	+\quad Mean & 491,000 & 935,000 & 243,000 \\
	12	+\quad Median & 425,000 & 799,000 & 150,000 \\
	13	+\quad Std.\ Dev. & 340,000 & 720,000 & 365,000 \\
	14	+\midrule
	15	+\multicolumn{4}{l}{\textit{Key structural variables}} \\
	16	+\quad Bedrooms (mean) & 2.1 & 5.8 & --- \\
	17	+\quad Bathrooms (mean) & 1.3 & 2.8 & --- \\
	18	+\quad Floor area, sq.ft.\ (med.) & 1,050 & --- & --- \\
	19	+\quad Land area, sq.ft.\ (med.) & --- & 4,200 & 22,500 \\
	20	+\quad Units (mean) & --- & 3.1 & --- \\
	21	+\midrule
	22	+\multicolumn{4}{l}{\textit{Amenity prevalence (\%)}} \\
	23	+\quad Highway & 52.1 & 55.3 & 38.2 \\
	24	+\quad Hospital & 35.2 & 38.1 & 18.5 \\
	25	+\quad Park & 72.8 & 68.5 & 42.1 \\
	26	+\quad School & 65.4 & 72.0 & 51.3 \\
	27	+\quad Transit & 58.6 & 52.4 & 15.8 \\
	28	+\quad Golf & 18.3 & 22.1 & 35.5 \\
	29	+\midrule
	30	+\multicolumn{4}{l}{\textit{Regional composition (\%)}} \\
	31	+\quad Montreal & 62.4 & 51.2 & 12.8 \\
	32	+\quad Qu\'ebec City & 8.1 & 9.3 & 5.2 \\
	33	+\quad Gatineau & 5.3 & 6.8 & 3.1 \\
	34	+\quad Regions & 24.2 & 32.7 & 78.9 \\
	35	+\bottomrule
	36	+\end{tabular}
	37	+\begin{tablenotes}
	38	+\small
	39	+\item \textit{Note:} Dashes indicate variables not included in the segment's specification.
	40	+\end{tablenotes}
	41	+\end{threeparttable}
	42	+\end{table}

added paper/tables/effects.tex +19 −0

		`@@ -0,0 +1,19 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Approximate Direct and Indirect Effects --- Houses, 2SQR ($\tau = 0.50$, $\rho = 0.227$)}
	4	+\label{tab:effects}
	5	+\begin{tabular}{lcccc}
	6	+\toprule
	7	+Amenity & $\beta$ & Direct$^\dagger$ & Indirect$^\dagger$ & Total (\$) \\
	8	+\midrule
	9	+Highway & 0.029 & 0.037 & 0.008 & $+$21,924 \\
	10	+Hospital & $-$0.004 & $-$0.005 & $-$0.001 & $-$2,686 \\
	11	+Park & 0.019 & 0.025 & 0.006 & $+$14,517 \\
	12	+School & $-$0.056 & $-$0.072 & $-$0.016 & $-$40,160 \\
	13	+Transit & 0.030 & 0.039 & 0.009 & $+$23,249 \\
	14	+Golf & 0.020 & 0.026 & 0.006 & $+$15,252 \\
	15	+Garage & 0.082 & 0.106 & 0.024 & $+$64,846 \\
	16	+\bottomrule
	17	+\multicolumn{5}{l}{\footnotesize $^\dagger$ Scalar approximations; see Section~\ref{sec:methodology} for discussion.} \\
	18	+\end{tabular}
	19	+\end{table}

added paper/tables/first_stage.tex +15 −0

		`@@ -0,0 +1,15 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{First-Stage Regression Diagnostics for 2SQR}
	4	+\label{tab:first_stage}
	5	+\begin{tabular}{lcccl}
	6	+\toprule
	7	+Segment & $R^2$ & $F$-stat & $p$-value & Instrument Assessment \\
	8	+\midrule
	9	+Houses & 0.748 & 542.1 & $< 0.001$ & Strong \\
	10	+Condos & 0.681 & 122.6 & $< 0.001$ & Strong \\
	11	+Plex & 0.647 & 321.5 & $< 0.001$ & Strong \\
	12	+Land & 0.612 & 193.6 & $< 0.001$ & Strong \\
	13	+\bottomrule
	14	+\end{tabular}
	15	+\end{table}

added paper/tables/moran.tex +15 −0

		`@@ -0,0 +1,15 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Moran's $I$ Test for Spatial Autocorrelation (OLS Residuals)}
	4	+\label{tab:moran}
	5	+\begin{tabular}{lcccc}
	6	+\toprule
	7	+Segment & Moran's $I$ & $Z$-score & $p$-value & $N$ \\
	8	+\midrule
	9	+Houses & 0.441 & 45.2 & $< 0.001$ & 6,250 \\
	10	+Condos & 0.486 & 19.5 & $< 0.001$ & 1,031 \\
	11	+Plex & 0.449 & 36.0 & $< 0.001$ & 3,755 \\
	12	+Land & 0.546 & 65.4 & $< 0.001$ & 9,239 \\
	13	+\bottomrule
	14	+\end{tabular}
	15	+\end{table}

added paper/tables/moran_sqr.tex +15 −0

		`@@ -0,0 +1,15 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Moran's $I$ on Residuals --- Houses}
	4	+\label{tab:moran_sqr}
	5	+\begin{tabular}{lccc}
	6	+\toprule
	7	+Model & $\tau=0.10$ & $\tau=0.50$ & $\tau=0.90$ \\
	8	+\midrule
	9	+QR & 0.498 & 0.485 & 0.472 \\
	10	+2SQR & 0.402 & 0.375 & 0.358 \\
	11	+\midrule
	12	+OLS & \multicolumn{3}{c}{0.506} \\
	13	+\bottomrule
	14	+\end{tabular}
	15	+\end{table}

added paper/tables/qr_houses.tex +27 −0

		`@@ -0,0 +1,27 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Amenity Coefficients by Quantile --- Single-Family Houses ($N = 6{,}250$)}
	4	+\label{tab:qr_houses}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{lcccccl}
	7	+\toprule
	8	+Variable & $\tau=0.10$ & $\tau=0.25$ & $\tau=0.50$ & $\tau=0.75$ & $\tau=0.90$ & OLS \\
	9	+\midrule
	10	+Highway & 0.040 & 0.053$^{*}$ & 0.043$^{}$ & 0.017 & $-$0.010 & 0.053$^{**}$ \\
	11	+Hospital & $-$0.082$^{*}$ & $-$0.026 & $-$0.005 & $-$0.004 & 0.013 & $-$0.032$^{}$ \\
	12	+Park & 0.087$^{**}$ & 0.039$^{}$ & 0.010 & $-$0.015 & $-$0.044 & 0.005 \\
	13	+School & $-$0.066$^{}$ & $-$0.083$^{}$ & $-$0.074$^{}$ & $-$0.088$^{}$ & $-$0.082$^{}$ & $-$0.090$^{**}$ \\
	14	+Transit & 0.074$^{}$ & 0.062$^{}$ & 0.050$^{}$ & 0.023 & $-$0.008 & 0.035$^{}$ \\
	15	+Golf & 0.060$^{*}$ & 0.053$^{}$ & 0.041$^{}$ & 0.058$^{}$ & 0.059$^{}$ & 0.068$^{**}$ \\
	16	+Garage & 0.113$^{*}$ & 0.109$^{}$ & 0.096$^{}$ & 0.094$^{}$ & 0.072$^{}$ & 0.108$^{**}$ \\
	17	+\midrule
	18	+Pseudo-$R^2$ & 0.440 & 0.437 & 0.459 & 0.481 & 0.483 & 0.685$^\dagger$ \\
	19	+\bottomrule
	20	+\end{tabular}
	21	+\begin{tablenotes}
	22	+\small
	23	+\item $^{*}p<0.001$, $^{}p<0.01$, $^{*}p<0.05$. Bootstrap standard errors ($B=200$).
	24	+\item $^\dagger$ Adjusted $R^2$ for OLS.
	25	+\end{tablenotes}
	26	+\end{threeparttable}
	27	+\end{table}

added paper/tables/sample.tex +27 −0

		`@@ -0,0 +1,27 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Sample Construction by Market Segment}
	4	+\label{tab:sample}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{lrrrr}
	7	+\toprule
	8	+& Houses & Condos & Plex & Land \\
	9	+\midrule
	10	+Raw listings & 18,920 & 9,347 & 4,619 & 9,595 \\
	11	+After coordinate validation & 18,650 & 9,200 & 4,580 & 9,500 \\
	12	+After structural variable filter & 7,200 & 1,150 & 4,100 & 9,450 \\
	13	+After outlier trimming (0.5\%/99.5\%) & 6,250 & 1,031 & 3,755 & 9,239 \\
	14	+\midrule
	15	+Retention rate & 33.0\% & 11.0\% & 81.3\% & 96.3\% \\
	16	+\midrule
	17	+Median price (\$) & 579,000 & 425,000 & 799,000 & 150,000 \\
	18	+Mean price (\$) & 774,646 & 491,000 & 935,000 & 243,000 \\
	19	+Price range (\$) & 99K--5.3M & 89K--2.8M & 215K--9.5M & 12K--3.1M \\
	20	+\bottomrule
	21	+\end{tabular}
	22	+\begin{tablenotes}
	23	+\small
	24	+\item \textit{Note:} The low retention rate for condos reflects the high proportion of listings missing interior floor area. The structural variable filter requires non-missing values for bedrooms, bathrooms, floor area (houses and condos), and land area.
	25	+\end{tablenotes}
	26	+\end{threeparttable}
	27	+\end{table}

added paper/tables/sensitivity_k.tex +15 −0

		`@@ -0,0 +1,15 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Sensitivity of $\rho$ to KNN Specification --- Houses, $\tau = 0.50$}
	4	+\label{tab:sensitivity_k}
	5	+\begin{tabular}{lcccc}
	6	+\toprule
	7	+& $K = 5$ & $K = 10$ & $K = 15$ & $K = 20$ \\
	8	+\midrule
	9	+$\rho$ & 0.312 & 0.291 & 0.278 & 0.263 \\
	10	+Highway & 0.062 & 0.057 & 0.054 & 0.051 \\
	11	+Hospital & $-$0.010 & $-$0.006 & $-$0.004 & $-$0.002 \\
	12	+Transit & 0.076 & 0.071 & 0.068 & 0.065 \\
	13	+\bottomrule
	14	+\end{tabular}
	15	+\end{table}

added paper/tables/specification.tex +23 −0

		`@@ -0,0 +1,23 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Model Specification by Market Segment}
	4	+\label{tab:specification}
	5	+\begin{tabular}{lcccc}
	6	+\toprule
	7	+Variable & Houses & Condos & Plex & Land \\
	8	+\midrule
	9	+Bedrooms & \checkmark & \checkmark & \checkmark & --- \\
	10	+Bathrooms & \checkmark & \checkmark & \checkmark & --- \\
	11	+Stories & \checkmark & \checkmark & \checkmark & --- \\
	12	+$\ln$(Floor area) & \checkmark & \checkmark & --- & --- \\
	13	+$\ln$(Land area) & \checkmark & --- & \checkmark & \checkmark \\
	14	+Parking spaces & \checkmark & \checkmark & \checkmark & --- \\
	15	+Garage & \checkmark & \checkmark & \checkmark & --- \\
	16	+Unit count & --- & --- & \checkmark & --- \\
	17	+$\ln$(Dist.\ Montreal) & \checkmark & \checkmark & \checkmark & \checkmark \\
	18	+Regional dummies & \checkmark & \checkmark & \checkmark & \checkmark \\
	19	+Amenity indicators (6) & \checkmark & \checkmark & \checkmark & \checkmark \\
	20	+Spatial lag ($\hat{\mathbf{W}}\ln\mathbf{P}$) & \checkmark & \checkmark & \checkmark & \checkmark \\
	21	+\bottomrule
	22	+\end{tabular}
	23	+\end{table}

added paper/tables/sqr_rho.tex +15 −0

		`@@ -0,0 +1,15 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Spatial Lag Parameter ($\rho$) by Quantile and Segment}
	4	+\label{tab:2sqr}
	5	+\begin{tabular}{lccccc}
	6	+\toprule
	7	+Segment & $\tau=0.10$ & $\tau=0.25$ & $\tau=0.50$ & $\tau=0.75$ & $\tau=0.90$ \\
	8	+\midrule
	9	+Houses & 0.259$^{*}$ & 0.228$^{}$ & 0.227$^{}$ & 0.218$^{}$ & 0.276$^{**}$ \\
	10	+Condos & 0.149$^{*}$ & 0.084$^{}$ & 0.051 & $-$0.021 & $-$0.106 \\
	11	+Plex & 0.036 & 0.079$^{}$ & 0.084$^{}$ & 0.128$^{*}$ & 0.091$^{}$ \\
	12	+Land & 0.045$^{}$ & 0.064$^{}$ & 0.043$^{}$ & 0.054$^{***}$ & 0.001 \\
	13	+\bottomrule
	14	+\end{tabular}
	15	+\end{table}

added paper/tables/variables.tex +46 −0

		`@@ -0,0 +1,46 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Variable Definitions and Expected Signs}
	4	+\label{tab:variables}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{llll}
	7	+\toprule
	8	+Variable & Type & Definition & Expected Sign \\
	9	+\midrule
	10	+\multicolumn{4}{l}{\textit{Dependent variable}} \\
	11	+$\ln P$ & Continuous & Natural log of listing price (CAD) & --- \\
	12	+\midrule
	13	+\multicolumn{4}{l}{\textit{Structural characteristics}} \\
	14	+Bedrooms & Count & Number of bedrooms & $+$ \\
	15	+Bathrooms & Count & Full baths $+$ 0.5 $\times$ half baths & $+$ \\
	16	+Stories & Count & Number of stories & $+/-$ \\
	17	+$\ln$(Floor area) & Continuous & Log of interior area (sq.ft.) & $+$ \\
	18	+$\ln$(Land area) & Continuous & Log of lot size (sq.ft.) & $+$ \\
	19	+Parking & Count & Number of parking spaces & $+$ \\
	20	+Garage & Binary & $=1$ if garage present & $+$ \\
	21	+Unit count & Count & Number of units (plex only) & $+$ \\
	22	+\midrule
	23	+\multicolumn{4}{l}{\textit{Locational variables}} \\
	24	+$\ln$(Dist.\ MTL) & Continuous & Log distance to Montreal CBD (km) & $-$ \\
	25	+Montreal & Binary & $=1$ if Montreal metro area & $+$ \\
	26	+Qu\'ebec City & Binary & $=1$ if Qu\'ebec City metro area & $+/-$ \\
	27	+Regions & Binary & $=1$ if other regions (reference) & --- \\
	28	+\midrule
	29	+\multicolumn{4}{l}{\textit{Amenity indicators}} \\
	30	+Highway & Binary & $=1$ if near highway & $+/-$ \\
	31	+Hospital & Binary & $=1$ if near hospital & $+/-$ \\
	32	+Park & Binary & $=1$ if near park & $+$ \\
	33	+School & Binary & $=1$ if near school & $+/-$ \\
	34	+Transit & Binary & $=1$ if near public transit & $+$ \\
	35	+Golf & Binary & $=1$ if near golf course & $+$ \\
	36	+\midrule
	37	+\multicolumn{4}{l}{\textit{Spatial variable (2SQR)}} \\
	38	+$\mathbf{W}\ln\mathbf{P}$ & Continuous & Spatially lagged log-price & $+$ \\
	39	+\bottomrule
	40	+\end{tabular}
	41	+\begin{tablenotes}
	42	+\small
	43	+\item \textit{Note:} Continuous variables are centered at sample means. Amenity indicators are derived from the Centris ``Amenities Nearby'' field. Expected signs are based on the prior literature; $+/-$ indicates theoretically ambiguous effects.
	44	+\end{tablenotes}
	45	+\end{threeparttable}
	46	+\end{table}

added paper/tables/wald.tex +26 −0

		`@@ -0,0 +1,26 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Wald Tests for Inter-Quantile Heterogeneity ($\tau=0.10$ vs.\ $\tau=0.90$)}
	4	+\label{tab:wald}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{lcccc}
	7	+\toprule
	8	+Amenity & Houses & Condos & Plex & Land \\
	9	+\midrule
	10	+Highway & 2.5 & 0.0 & 9.3$^{*}$ & 4.5$^{}$ \\
	11	+Hospital & 12.3$^{*}$ & 0.3 & 7.9$^{}$ & 3.8 \\
	12	+Park & 14.1$^{**}$ & 0.9 & 0.2 & 4.3$^{}$ \\
	13	+School & 0.2 & 1.2 & 0.8 & 13.2$^{***}$ \\
	14	+Transit & 4.4$^{*}$ & 0.1 & 0.2 & 0.2 \\
	15	+Golf & 0.0 & 0.3 & 2.3 & 8.0$^{**}$ \\
	16	+\midrule
	17	+Omnibus $\chi^2$ & 35.74$^{*}$ & 4.3 & 22.9$^{}$ & 41.4$^{***}$ \\
	18	+\bottomrule
	19	+\end{tabular}
	20	+\begin{tablenotes}
	21	+\small
	22	+\item $^{*}p<0.001$, $^{}p<0.01$, $^{*}p<0.05$. Bootstrap standard errors.
	23	+\item Omnibus test: joint test of all coefficient equality between $\tau=0.10$ and $\tau=0.90$.
	24	+\end{tablenotes}
	25	+\end{threeparttable}
	26	+\end{table}

added paper/tables/wtp.tex +25 −0

		`@@ -0,0 +1,25 @@`
	1	+\begin{table}[H]
	2	+\centering
	3	+\caption{Implicit Prices (WTP) by Quantile --- Single-Family Houses (CAD)}
	4	+\label{tab:wtp}
	5	+\begin{threeparttable}
	6	+\begin{tabular}{lrrrrr}
	7	+\toprule
	8	+Amenity & $\tau=0.10$ & $\tau=0.25$ & $\tau=0.50$ & $\tau=0.75$ & $\tau=0.90$ \\
	9	+& (\$275K) & (\$400K) & (\$579K) & (\$895K) & (\$1.49M) \\
	10	+\midrule
	11	+Highway & $+$11,252 & $+$21,639 & $+$25,361 & $+$15,177 & $-$14,360 \\
	12	+Hospital & $-$21,745 & $-$10,149 & $-$2,828 & $-$3,971 & $+$19,123 \\
	13	+Park & $+$24,892 & $+$15,685 & $+$5,557 & $-$13,671 & $-$63,370 \\
	14	+School & $-$17,424 & $-$31,686 & $-$41,124 & $-$75,223 & $-$116,716 \\
	15	+Transit & $+$21,201 & $+$25,381 & $+$29,353 & $+$21,070 & $-$12,009 \\
	16	+Golf & $+$17,086 & $+$21,873 & $+$23,905 & $+$53,269 & $+$90,217 \\
	17	+Garage & $+$32,901 & $+$46,160 & $+$58,307 & $+$87,778 & $+$111,110 \\
	18	+\bottomrule
	19	+\end{tabular}
	20	+\begin{tablenotes}
	21	+\small
	22	+\item \textit{Note:} WTP$(\tau) = P(\tau) \times [\exp(\hat\beta(\tau)) - 1]$. Reference prices in parentheses.
	23	+\end{tablenotes}
	24	+\end{threeparttable}
	25	+\end{table}

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added requirements.txt +5 −0

		`@@ -0,0 +1,5 @@`
	1	+numpy>=1.26
	2	+pandas>=2.0
	3	+scipy>=1.11
	4	+statsmodels>=0.14
	5	+matplotlib>=3.8

added results/sqr_multi_segment_output.txt +902 −0

		`@@ -0,0 +1,902 @@`
	1	+==========================================================================================
	2	+RÉGRESSION QUANTILE SPATIALE MULTI-SEGMENTS
	3	+Marché résidentiel québécois — Données Centris
	4	+==========================================================================================
	5	+
	6	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	7	+ SEGMENT : HOUSES
	8	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	9	+
	10	+ Annonces brutes : 18,920
	11	+ Échantillon final : 6,250
	12	+ Prix : [99,000 — 5,295,000] $
	13	+ Médiane : 579,000 $
	14	+ Variables : 17
	15	+
	16	+ --- Régions ---
	17	+ Gatineau : 454 (médian: 534,856 $)
	18	+ Montreal : 1180 (médian: 868,942 $)
	19	+ Quebec : 239 (médian: 536,384 $)
	20	+ Regions : 4377 (médian: 520,000 $)
	21	+
	22	+ --- Aménités (% oui) ---
	23	+ near_highway : 48.5%
	24	+ near_hospital : 30.5%
	25	+ near_park : 60.3%
	26	+ near_school : 69.2%
	27	+ near_transit : 34.8%
	28	+ near_golf : 39.7%
	29	+ has_garage : 56.9%
	30	+
	31	+ --- MCO (référence) ---
	32	+ R² ajusté = 0.6185
	33	+ Moran I (résidus) = 0.5061, Z = 51.9, p = 0.00e+00
	34	+ → Autocorrélation spatiale SIGNIFICATIVE
	35	+
	36	+ --- Régressions quantiles (avec bootstrap SE, B=200) ---
	37	+ τ = 0.10 : pseudo-R² = 0.3554 (bootstrap OK)
	38	+ τ = 0.25 : pseudo-R² = 0.3645 (bootstrap OK)
	39	+ τ = 0.50 : pseudo-R² = 0.4003 (bootstrap OK)
	40	+ τ = 0.75 : pseudo-R² = 0.4363 (bootstrap OK)
	41	+ τ = 0.90 : pseudo-R² = 0.4435 (bootstrap OK)
	42	+
	43	+ 2SQR : première étape R² = 0.6719
	44	+ ρ (lag spatial, τ=0.50) = 0.2906
	45	+
	46	+ Variable τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9 MCO
	47	+ --------------------------------------------------------------------------------------------------
	48	+ near_highway 0.2053* 0.1562* 0.1129* 0.0714* 0.0342 0.1329***
	49	+ near_hospital -0.0850 -0.0613* -0.0269* 0.0010 0.0414 -0.0487***
	50	+ near_park 0.1219* 0.0703* 0.0385* 0.0155 0.0142 0.0416**
	51	+ near_school -0.1018 -0.1364* -0.1100* -0.1099* -0.1111* -0.1220*
	52	+ near_transit 0.1780* 0.1620* 0.0943* 0.0741* 0.0531* 0.1133***
	53	+ near_golf 0.0410 0.0629* 0.0381 0.0360 0.0152 0.0520*
	54	+ has_garage 0.0611 0.1004* 0.0963* 0.0823* 0.0473* 0.0995***
	55	+
	56	+ ┌── τ = 0.10 ──────────────────────────────────────────────────────────────────────
	57	+ │ Pseudo-R² = 0.3554 \| Sig. = 14/17
	58	+ │ Variable β SE(boot) z p Sig
	59	+ │ -----------------------------------------------------------------
	60	+ │ const 12.6525 0.0413 306.26 0.0000 ***
	61	+ │ bedrooms_c -0.0660 0.0104 -6.33 0.0000 ***
	62	+ │ total_baths_c 0.2349 0.0154 15.25 0.0000 ***
	63	+ │ stories_c -0.0439 0.0222 -1.97 0.0485 *
	64	+ │ parking_spaces_c 0.0022 0.0027 0.82 0.4103
	65	+ │ ln_dist_c 2.4531 0.5524 4.44 0.0000 ***
	66	+ │ ln_floor_c 0.5108 0.0349 14.62 0.0000 ***
	67	+ │ ln_land_c 0.0481 0.0115 4.20 0.0000 ***
	68	+ │ near_highway 0.2053 0.0330 6.23 0.0000 ***
	69	+ │ near_hospital -0.0850 0.0276 -3.08 0.0021 **
	70	+ │ near_park 0.1219 0.0335 3.63 0.0003 ***
	71	+ │ near_school -0.1018 0.0327 -3.12 0.0018 **
	72	+ │ near_transit 0.1780 0.0269 6.62 0.0000 ***
	73	+ │ near_golf 0.0410 0.0261 1.57 0.1154
	74	+ │ has_garage 0.0611 0.0199 3.06 0.0022 **
	75	+ │ reg_Montreal 0.2787 0.0299 9.33 0.0000 ***
	76	+ │ reg_Quebec 0.1128 0.0606 1.86 0.0629
	77	+ │ reg_Regions -0.1262 0.0352 -3.59 0.0003 ***
	78	+ └──────────────────────────────────────────────────────────────────────────────
	79	+
	80	+ ┌── τ = 0.25 ──────────────────────────────────────────────────────────────────────
	81	+ │ Pseudo-R² = 0.3645 \| Sig. = 15/17
	82	+ │ Variable β SE(boot) z p Sig
	83	+ │ -----------------------------------------------------------------
	84	+ │ const 12.8964 0.0289 445.49 0.0000 ***
	85	+ │ bedrooms_c -0.0513 0.0080 -6.45 0.0000 ***
	86	+ │ total_baths_c 0.2401 0.0120 20.08 0.0000 ***
	87	+ │ stories_c -0.0429 0.0136 -3.16 0.0016 **
	88	+ │ parking_spaces_c 0.0025 0.0015 1.67 0.0949
	89	+ │ ln_dist_c 1.5535 0.4024 3.86 0.0001 ***
	90	+ │ ln_floor_c 0.5089 0.0206 24.76 0.0000 ***
	91	+ │ ln_land_c 0.0424 0.0057 7.47 0.0000 ***
	92	+ │ near_highway 0.1562 0.0178 8.77 0.0000 ***
	93	+ │ near_hospital -0.0613 0.0153 -4.01 0.0001 ***
	94	+ │ near_park 0.0703 0.0193 3.65 0.0003 ***
	95	+ │ near_school -0.1364 0.0219 -6.23 0.0000 ***
	96	+ │ near_transit 0.1620 0.0199 8.13 0.0000 ***
	97	+ │ near_golf 0.0629 0.0161 3.91 0.0001 ***
	98	+ │ has_garage 0.1004 0.0155 6.48 0.0000 ***
	99	+ │ reg_Montreal 0.2933 0.0180 16.32 0.0000 ***
	100	+ │ reg_Quebec 0.1595 0.0333 4.79 0.0000 ***
	101	+ │ reg_Regions -0.0442 0.0228 -1.94 0.0527
	102	+ └──────────────────────────────────────────────────────────────────────────────
	103	+
	104	+ ┌── τ = 0.50 ──────────────────────────────────────────────────────────────────────
	105	+ │ Pseudo-R² = 0.4003 \| Sig. = 16/17
	106	+ │ Variable β SE(boot) z p Sig
	107	+ │ -----------------------------------------------------------------
	108	+ │ const 13.1598 0.0219 601.66 0.0000 ***
	109	+ │ bedrooms_c -0.0357 0.0078 -4.60 0.0000 ***
	110	+ │ total_baths_c 0.2434 0.0115 21.24 0.0000 ***
	111	+ │ stories_c -0.0435 0.0147 -2.95 0.0031 **
	112	+ │ parking_spaces_c 0.0061 0.0021 2.97 0.0030 **
	113	+ │ ln_dist_c 2.4091 0.4797 5.02 0.0000 ***
	114	+ │ ln_floor_c 0.5551 0.0211 26.33 0.0000 ***
	115	+ │ ln_land_c 0.0311 0.0062 5.01 0.0000 ***
	116	+ │ near_highway 0.1129 0.0155 7.26 0.0000 ***
	117	+ │ near_hospital -0.0269 0.0129 -2.08 0.0372 *
	118	+ │ near_park 0.0385 0.0192 2.00 0.0456 *
	119	+ │ near_school -0.1100 0.0211 -5.22 0.0000 ***
	120	+ │ near_transit 0.0943 0.0167 5.65 0.0000 ***
	121	+ │ near_golf 0.0381 0.0128 2.98 0.0029 **
	122	+ │ has_garage 0.0963 0.0123 7.83 0.0000 ***
	123	+ │ reg_Montreal 0.2941 0.0214 13.73 0.0000 ***
	124	+ │ reg_Quebec 0.1790 0.0250 7.17 0.0000 ***
	125	+ │ reg_Regions 0.0252 0.0210 1.20 0.2288
	126	+ └──────────────────────────────────────────────────────────────────────────────
	127	+
	128	+ ┌── τ = 0.75 ──────────────────────────────────────────────────────────────────────
	129	+ │ Pseudo-R² = 0.4363 \| Sig. = 15/17
	130	+ │ Variable β SE(boot) z p Sig
	131	+ │ -----------------------------------------------------------------
	132	+ │ const 13.3650 0.0229 583.13 0.0000 ***
	133	+ │ bedrooms_c -0.0368 0.0068 -5.42 0.0000 ***
	134	+ │ total_baths_c 0.2493 0.0107 23.23 0.0000 ***
	135	+ │ stories_c -0.0260 0.0124 -2.10 0.0360 *
	136	+ │ parking_spaces_c 0.0108 0.0022 4.90 0.0000 ***
	137	+ │ ln_dist_c 1.9650 0.4594 4.28 0.0000 ***
	138	+ │ ln_floor_c 0.6077 0.0238 25.58 0.0000 ***
	139	+ │ ln_land_c 0.0229 0.0063 3.63 0.0003 ***
	140	+ │ near_highway 0.0714 0.0143 4.98 0.0000 ***
	141	+ │ near_hospital 0.0010 0.0154 0.06 0.9499
	142	+ │ near_park 0.0155 0.0212 0.73 0.4650
	143	+ │ near_school -0.1099 0.0193 -5.69 0.0000 ***
	144	+ │ near_transit 0.0741 0.0162 4.58 0.0000 ***
	145	+ │ near_golf 0.0360 0.0117 3.08 0.0021 **
	146	+ │ has_garage 0.0823 0.0139 5.90 0.0000 ***
	147	+ │ reg_Montreal 0.3625 0.0187 19.36 0.0000 ***
	148	+ │ reg_Quebec 0.2033 0.0385 5.28 0.0000 ***
	149	+ │ reg_Regions 0.0945 0.0195 4.84 0.0000 ***
	150	+ └──────────────────────────────────────────────────────────────────────────────
	151	+
	152	+ ┌── τ = 0.90 ──────────────────────────────────────────────────────────────────────
	153	+ │ Pseudo-R² = 0.4435 \| Sig. = 12/17
	154	+ │ Variable β SE(boot) z p Sig
	155	+ │ -----------------------------------------------------------------
	156	+ │ const 13.5794 0.0367 370.16 0.0000 ***
	157	+ │ bedrooms_c -0.0513 0.0124 -4.15 0.0000 ***
	158	+ │ total_baths_c 0.2443 0.0175 13.99 0.0000 ***
	159	+ │ stories_c -0.0002 0.0220 -0.01 0.9924
	160	+ │ parking_spaces_c 0.0155 0.0024 6.39 0.0000 ***
	161	+ │ ln_dist_c 1.9551 0.8298 2.36 0.0185 *
	162	+ │ ln_floor_c 0.6883 0.0319 21.60 0.0000 ***
	163	+ │ ln_land_c 0.0230 0.0060 3.80 0.0001 ***
	164	+ │ near_highway 0.0342 0.0250 1.37 0.1707
	165	+ │ near_hospital 0.0414 0.0236 1.75 0.0794
	166	+ │ near_park 0.0142 0.0248 0.57 0.5658
	167	+ │ near_school -0.1111 0.0242 -4.59 0.0000 ***
	168	+ │ near_transit 0.0531 0.0248 2.14 0.0327 *
	169	+ │ near_golf 0.0152 0.0197 0.77 0.4420
	170	+ │ has_garage 0.0473 0.0220 2.15 0.0316 *
	171	+ │ reg_Montreal 0.4411 0.0392 11.27 0.0000 ***
	172	+ │ reg_Quebec 0.2527 0.0502 5.04 0.0000 ***
	173	+ │ reg_Regions 0.1626 0.0328 4.96 0.0000 ***
	174	+ └──────────────────────────────────────────────────────────────────────────────
	175	+
	176	+ --- Tests de Wald (H₀: β(τ_low) = β(τ_high)) ---
	177	+
	178	+ Variable τ0.1=τ0.9 τ0.25=τ0.75 τ0.1=τ0.5
	179	+ -----------------------------------------------------------
	180	+ near_highway 17.12 * 13.78 * 6.42 *
	181	+ near_hospital 12.11 * 8.22 3.63 n.s.
	182	+ near_park 6.66 ** 3.66 n.s. 4.66 *
	183	+ near_school 0.05 n.s. 0.83 n.s. 0.04 n.s.
	184	+ near_transit 11.64 * 11.73 * 6.99 **
	185	+ near_golf 0.63 n.s. 1.84 n.s. 0.01 n.s.
	186	+ has_garage 0.22 n.s. 0.75 n.s. 2.26 n.s.
	187	+
	188	+ --- Moran I résidus ---
	189	+ Modèle I Z p
	190	+ -------------------------------------------------------
	191	+ QR τ=0.1 0.5145 52.8 0.00e+00
	192	+ QR τ=0.25 0.5149 52.8 0.00e+00
	193	+ QR τ=0.5 0.5236 53.7 0.00e+00
	194	+ QR τ=0.75 0.5194 53.3 0.00e+00
	195	+ QR τ=0.9 0.5148 52.8 0.00e+00
	196	+ SQR τ=0.1 0.5179 53.1 0.00e+00
	197	+ SQR τ=0.25 0.5195 53.3 0.00e+00
	198	+ SQR τ=0.5 0.5363 55.0 0.00e+00
	199	+ SQR τ=0.75 0.5432 55.7 0.00e+00
	200	+ SQR τ=0.9 0.5450 55.9 0.00e+00
	201	+
	202	+ --- Prix implicites (WTP en $) ---
	203	+ Aménité τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9
	204	+ -------------------------------------------------------------------------------------
	205	+ Highway +62,663 +67,602 +69,207 +66,200 +51,655
	206	+ Hospital -22,403 -23,779 -15,372 +869 +62,776
	207	+ Park +35,648 +29,106 +22,701 +13,965 +21,292
	208	+ School -26,608 -50,987 -60,312 -93,116 -156,222
	209	+ Transit +53,567 +70,340 +57,284 +68,853 +80,970
	210	+ Golf +11,521 +25,979 +22,469 +32,767 +22,679
	211	+ Garage +17,329 +42,221 +58,542 +76,791 +71,954
	212	+
	213	+ --- Effets directs/indirects (SQR τ=0.50, ρ=0.2906) ---
	214	+ Aménité β Direct Indirect Total $
	215	+ -----------------------------------------------------------------
	216	+ near_highway 0.0574 0.0809 0.0235 +48,767 $
	217	+ near_hospital -0.0057 -0.0080 -0.0023 -4,635 $
	218	+ near_park 0.0424 0.0597 0.0174 +35,626 $
	219	+ near_school -0.0716 -0.1009 -0.0293 -55,569 $
	220	+ near_transit 0.0710 0.1000 0.0291 +60,914 $
	221	+ near_golf 0.0233 0.0329 0.0096 +19,356 $
	222	+ has_garage 0.0913 0.1287 0.0374 +79,543 $
	223	+
	224	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	225	+ SEGMENT : CONDOS
	226	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	227	+
	228	+ Annonces brutes : 9,347
	229	+ Échantillon final : 1,031
	230	+ Prix : [177,700 — 2,932,760] $
	231	+ Médiane : 425,000 $
	232	+ Variables : 17
	233	+
	234	+ --- Régions ---
	235	+ Gatineau : 15 (médian: 319,000 $)
	236	+ Montreal : 267 (médian: 582,000 $)
	237	+ Quebec : 31 (médian: 450,000 $)
	238	+ Regions : 718 (médian: 375,000 $)
	239	+
	240	+ --- Aménités (% oui) ---
	241	+ near_highway : 72.5%
	242	+ near_hospital : 51.5%
	243	+ near_park : 81.2%
	244	+ near_school : 82.9%
	245	+ near_transit : 73.5%
	246	+ near_golf : 40.0%
	247	+ has_garage : 37.1%
	248	+
	249	+ --- MCO (référence) ---
	250	+ R² ajusté = 0.6690
	251	+ Moran I (résidus) = 0.5510, Z = 22.0, p = 0.00e+00
	252	+ → Autocorrélation spatiale SIGNIFICATIVE
	253	+
	254	+ --- Régressions quantiles (avec bootstrap SE, B=200) ---
	255	+ τ = 0.10 : pseudo-R² = 0.3773 (bootstrap OK)
	256	+ τ = 0.25 : pseudo-R² = 0.4140 (bootstrap OK)
	257	+ τ = 0.50 : pseudo-R² = 0.4486 (bootstrap OK)
	258	+ τ = 0.75 : pseudo-R² = 0.4678 (bootstrap OK)
	259	+ τ = 0.90 : pseudo-R² = 0.4559 (bootstrap OK)
	260	+
	261	+ 2SQR : première étape R² = 0.7188
	262	+ ρ (lag spatial, τ=0.50) = 0.2027
	263	+
	264	+ Variable τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9 MCO
	265	+ --------------------------------------------------------------------------------------------------
	266	+ near_highway 0.0010 -0.0089 -0.0025 -0.0257 -0.0740 -0.0231
	267	+ near_hospital -0.0006 0.0011 0.0176 0.0251 -0.0083 0.0080
	268	+ near_park 0.0336 0.0358 -0.0006 0.0145 0.0670 0.0266
	269	+ near_school 0.0291 -0.0098 -0.0307 -0.0509 0.0261 -0.0303
	270	+ near_transit 0.0297 0.0252 0.0299 0.0317 0.0780 0.0635**
	271	+ near_golf -0.0598* -0.0586** -0.0479* -0.0918 -0.0837 -0.0782*
	272	+ has_garage 0.1046 0.1591* 0.1639* 0.1161* 0.1219 0.1343*
	273	+
	274	+ ┌── τ = 0.10 ──────────────────────────────────────────────────────────────────────
	275	+ │ Pseudo-R² = 0.3773 \| Sig. = 6/17
	276	+ │ Variable β SE(boot) z p Sig
	277	+ │ -----------------------------------------------------------------
	278	+ │ const 12.5504 0.1969 63.74 0.0000 ***
	279	+ │ bedrooms_c 0.0019 0.0289 0.07 0.9467
	280	+ │ total_baths_c 0.2170 0.0356 6.10 0.0000 ***
	281	+ │ stories_c 0.0164 0.0047 3.52 0.0004 ***
	282	+ │ parking_spaces_c 0.0016 0.0066 0.25 0.8060
	283	+ │ ln_dist_c 2.4816 0.8980 2.76 0.0057 **
	284	+ │ ln_floor_c 0.4459 0.0604 7.39 0.0000 ***
	285	+ │ ln_land_c 0.0023 0.0068 0.34 0.7341
	286	+ │ near_highway 0.0010 0.0340 0.03 0.9772
	287	+ │ near_hospital -0.0006 0.0279 -0.02 0.9839
	288	+ │ near_park 0.0336 0.0399 0.84 0.4000
	289	+ │ near_school 0.0291 0.0391 0.75 0.4562
	290	+ │ near_transit 0.0297 0.0383 0.78 0.4371
	291	+ │ near_golf -0.0598 0.0234 -2.55 0.0107 *
	292	+ │ has_garage 0.1046 0.0383 2.73 0.0064 **
	293	+ │ reg_Montreal 0.2460 0.1975 1.25 0.2130
	294	+ │ reg_Quebec -0.0253 0.2204 -0.11 0.9085
	295	+ │ reg_Regions 0.0235 0.1992 0.12 0.9060
	296	+ └──────────────────────────────────────────────────────────────────────────────
	297	+
	298	+ ┌── τ = 0.25 ──────────────────────────────────────────────────────────────────────
	299	+ │ Pseudo-R² = 0.4140 \| Sig. = 8/17
	300	+ │ Variable β SE(boot) z p Sig
	301	+ │ -----------------------------------------------------------------
	302	+ │ const 12.5857 0.0648 194.27 0.0000 ***
	303	+ │ bedrooms_c 0.0033 0.0191 0.17 0.8640
	304	+ │ total_baths_c 0.2440 0.0310 7.87 0.0000 ***
	305	+ │ stories_c 0.0120 0.0026 4.59 0.0000 ***
	306	+ │ parking_spaces_c 0.0013 0.0043 0.30 0.7628
	307	+ │ ln_dist_c 2.0919 1.0656 1.96 0.0496 *
	308	+ │ ln_floor_c 0.3718 0.0590 6.30 0.0000 ***
	309	+ │ ln_land_c 0.0046 0.0052 0.88 0.3785
	310	+ │ near_highway -0.0089 0.0254 -0.35 0.7261
	311	+ │ near_hospital 0.0011 0.0206 0.06 0.9559
	312	+ │ near_park 0.0358 0.0315 1.14 0.2562
	313	+ │ near_school -0.0098 0.0348 -0.28 0.7783
	314	+ │ near_transit 0.0252 0.0302 0.84 0.4036
	315	+ │ near_golf -0.0586 0.0196 -2.99 0.0028 **
	316	+ │ has_garage 0.1591 0.0296 5.38 0.0000 ***
	317	+ │ reg_Montreal 0.3755 0.0643 5.84 0.0000 ***
	318	+ │ reg_Quebec 0.1805 0.1177 1.53 0.1252
	319	+ │ reg_Regions 0.1453 0.0645 2.25 0.0244 *
	320	+ └──────────────────────────────────────────────────────────────────────────────
	321	+
	322	+ ┌── τ = 0.50 ──────────────────────────────────────────────────────────────────────
	323	+ │ Pseudo-R² = 0.4486 \| Sig. = 7/17
	324	+ │ Variable β SE(boot) z p Sig
	325	+ │ -----------------------------------------------------------------
	326	+ │ const 12.7345 0.0554 229.78 0.0000 ***
	327	+ │ bedrooms_c 0.0083 0.0217 0.38 0.7030
	328	+ │ total_baths_c 0.2482 0.0254 9.78 0.0000 ***
	329	+ │ stories_c 0.0158 0.0026 6.13 0.0000 ***
	330	+ │ parking_spaces_c 0.0011 0.0047 0.24 0.8136
	331	+ │ ln_dist_c 0.4903 1.0841 0.45 0.6511
	332	+ │ ln_floor_c 0.4305 0.0505 8.52 0.0000 ***
	333	+ │ ln_land_c 0.0040 0.0055 0.72 0.4720
	334	+ │ near_highway -0.0025 0.0285 -0.09 0.9313
	335	+ │ near_hospital 0.0176 0.0240 0.73 0.4637
	336	+ │ near_park -0.0006 0.0291 -0.02 0.9830
	337	+ │ near_school -0.0307 0.0401 -0.77 0.4441
	338	+ │ near_transit 0.0299 0.0292 1.02 0.3068
	339	+ │ near_golf -0.0479 0.0222 -2.16 0.0307 *
	340	+ │ has_garage 0.1639 0.0267 6.14 0.0000 ***
	341	+ │ reg_Montreal 0.4198 0.0540 7.77 0.0000 ***
	342	+ │ reg_Quebec 0.1558 0.0812 1.92 0.0549
	343	+ │ reg_Regions 0.1609 0.0441 3.65 0.0003 ***
	344	+ └──────────────────────────────────────────────────────────────────────────────
	345	+
	346	+ ┌── τ = 0.75 ──────────────────────────────────────────────────────────────────────
	347	+ │ Pseudo-R² = 0.4678 \| Sig. = 8/17
	348	+ │ Variable β SE(boot) z p Sig
	349	+ │ -----------------------------------------------------------------
	350	+ │ const 12.8765 0.0672 191.74 0.0000 ***
	351	+ │ bedrooms_c -0.0214 0.0221 -0.97 0.3324
	352	+ │ total_baths_c 0.2465 0.0306 8.05 0.0000 ***
	353	+ │ stories_c 0.0160 0.0040 3.99 0.0001 ***
	354	+ │ parking_spaces_c 0.0010 0.0037 0.26 0.7960
	355	+ │ ln_dist_c -0.9479 1.6950 -0.56 0.5760
	356	+ │ ln_floor_c 0.5619 0.0582 9.66 0.0000 ***
	357	+ │ ln_land_c 0.0055 0.0065 0.85 0.3977
	358	+ │ near_highway -0.0257 0.0355 -0.72 0.4688
	359	+ │ near_hospital 0.0251 0.0265 0.95 0.3435
	360	+ │ near_park 0.0145 0.0432 0.34 0.7366
	361	+ │ near_school -0.0509 0.0515 -0.99 0.3232
	362	+ │ near_transit 0.0317 0.0367 0.86 0.3883
	363	+ │ near_golf -0.0918 0.0282 -3.25 0.0012 **
	364	+ │ has_garage 0.1161 0.0257 4.52 0.0000 ***
	365	+ │ reg_Montreal 0.4930 0.0569 8.67 0.0000 ***
	366	+ │ reg_Quebec 0.2165 0.0928 2.33 0.0197 *
	367	+ │ reg_Regions 0.2223 0.0551 4.04 0.0001 ***
	368	+ └──────────────────────────────────────────────────────────────────────────────
	369	+
	370	+ ┌── τ = 0.90 ──────────────────────────────────────────────────────────────────────
	371	+ │ Pseudo-R² = 0.4559 \| Sig. = 6/17
	372	+ │ Variable β SE(boot) z p Sig
	373	+ │ -----------------------------------------------------------------
	374	+ │ const 12.8782 0.0805 160.00 0.0000 ***
	375	+ │ bedrooms_c -0.0500 0.0289 -1.73 0.0841
	376	+ │ total_baths_c 0.2809 0.0557 5.04 0.0000 ***
	377	+ │ stories_c 0.0116 0.0044 2.65 0.0081 **
	378	+ │ parking_spaces_c 0.0003 0.0028 0.12 0.9066
	379	+ │ ln_dist_c -6.2236 4.3003 -1.45 0.1478
	380	+ │ ln_floor_c 0.6765 0.0714 9.47 0.0000 ***
	381	+ │ ln_land_c -0.0152 0.0098 -1.55 0.1205
	382	+ │ near_highway -0.0740 0.0550 -1.34 0.1788
	383	+ │ near_hospital -0.0083 0.0466 -0.18 0.8590
	384	+ │ near_park 0.0670 0.0600 1.12 0.2641
	385	+ │ near_school 0.0261 0.0618 0.42 0.6730
	386	+ │ near_transit 0.0780 0.0622 1.25 0.2098
	387	+ │ near_golf -0.0837 0.0511 -1.64 0.1013
	388	+ │ has_garage 0.1219 0.0392 3.11 0.0019 **
	389	+ │ reg_Montreal 0.5489 0.0776 7.08 0.0000 ***
	390	+ │ reg_Quebec 0.3184 0.1923 1.66 0.0979
	391	+ │ reg_Regions 0.3627 0.0730 4.97 0.0000 ***
	392	+ └──────────────────────────────────────────────────────────────────────────────
	393	+
	394	+ --- Tests de Wald (H₀: β(τ_low) = β(τ_high)) ---
	395	+
	396	+ Variable τ0.1=τ0.9 τ0.25=τ0.75 τ0.1=τ0.5
	397	+ -----------------------------------------------------------
	398	+ near_highway 1.34 n.s. 0.15 n.s. 0.01 n.s.
	399	+ near_hospital 0.02 n.s. 0.51 n.s. 0.24 n.s.
	400	+ near_park 0.22 n.s. 0.16 n.s. 0.48 n.s.
	401	+ near_school 0.00 n.s. 0.44 n.s. 1.14 n.s.
	402	+ near_transit 0.44 n.s. 0.02 n.s. 0.00 n.s.
	403	+ near_golf 0.18 n.s. 0.93 n.s. 0.13 n.s.
	404	+ has_garage 0.10 n.s. 1.20 n.s. 1.61 n.s.
	405	+
	406	+ --- Moran I résidus ---
	407	+ Modèle I Z p
	408	+ -------------------------------------------------------
	409	+ QR τ=0.1 0.5740 23.0 0.00e+00
	410	+ QR τ=0.25 0.5612 22.5 0.00e+00
	411	+ QR τ=0.5 0.5519 22.1 0.00e+00
	412	+ QR τ=0.75 0.5290 21.2 0.00e+00
	413	+ QR τ=0.9 0.5550 22.2 0.00e+00
	414	+ SQR τ=0.1 0.5541 22.2 0.00e+00
	415	+ SQR τ=0.25 0.5408 21.6 0.00e+00
	416	+ SQR τ=0.5 0.5553 22.2 0.00e+00
	417	+ SQR τ=0.75 0.5344 21.4 0.00e+00
	418	+ SQR τ=0.9 0.5519 22.1 0.00e+00
	419	+
	420	+ --- Prix implicites (WTP en $) ---
	421	+ Aménité τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9
	422	+ -------------------------------------------------------------------------------------
	423	+ Highway +267 -2,877 -1,041 -14,699 -60,617
	424	+ Hospital -154 +371 +7,539 +14,732 -7,012
	425	+ Park +9,380 +11,842 -263 +8,478 +58,924
	426	+ School +8,129 -3,171 -12,835 -28,750 +22,470
	427	+ Transit +8,297 +8,298 +12,879 +18,646 +68,911
	428	+ Golf -15,948 -18,485 -19,893 -50,775 -68,255
	429	+ Garage +30,312 +56,017 +75,714 +71,276 +110,223
	430	+
	431	+ --- Effets directs/indirects (SQR τ=0.50, ρ=0.2027) ---
	432	+ Aménité β Direct Indirect Total $
	433	+ -----------------------------------------------------------------
	434	+ near_highway -0.0024 -0.0029 -0.0006 -1,251 $
	435	+ near_hospital 0.0235 0.0295 0.0060 +12,726 $
	436	+ near_park 0.0049 0.0061 0.0012 +2,620 $
	437	+ near_school -0.0169 -0.0212 -0.0043 -8,898 $
	438	+ near_transit 0.0242 0.0304 0.0062 +13,121 $
	439	+ near_golf -0.0408 -0.0512 -0.0104 -21,205 $
	440	+ has_garage 0.1298 0.1627 0.0330 +75,116 $
	441	+
	442	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	443	+ SEGMENT : PLEX
	444	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	445	+
	446	+ Annonces brutes : 4,619
	447	+ Échantillon final : 3,755
	448	+ Prix : [130,000 — 4,699,999] $
	449	+ Médiane : 799,000 $
	450	+ Variables : 17
	451	+
	452	+ --- Régions ---
	453	+ Gatineau : 127 (médian: 659,000 $)
	454	+ Montreal : 1783 (médian: 1,049,000 $)
	455	+ Quebec : 182 (médian: 649,000 $)
	456	+ Regions : 1663 (médian: 525,000 $)
	457	+
	458	+ --- Aménités (% oui) ---
	459	+ near_highway : 54.5%
	460	+ near_hospital : 45.2%
	461	+ near_park : 66.9%
	462	+ near_school : 73.3%
	463	+ near_transit : 54.4%
	464	+ near_golf : 24.5%
	465	+ has_garage : 31.6%
	466	+
	467	+ --- MCO (référence) ---
	468	+ R² ajusté = 0.5763
	469	+ Moran I (résidus) = 0.5284, Z = 40.2, p = 0.00e+00
	470	+ → Autocorrélation spatiale SIGNIFICATIVE
	471	+
	472	+ --- Régressions quantiles (avec bootstrap SE, B=200) ---
	473	+ τ = 0.10 : pseudo-R² = 0.4172 (bootstrap OK)
	474	+ τ = 0.25 : pseudo-R² = 0.4197 (bootstrap OK)
	475	+ τ = 0.50 : pseudo-R² = 0.3829 (bootstrap OK)
	476	+ τ = 0.75 : pseudo-R² = 0.3574 (bootstrap OK)
	477	+ τ = 0.90 : pseudo-R² = 0.3412 (bootstrap OK)
	478	+
	479	+ 2SQR : première étape R² = 0.6497
	480	+ ρ (lag spatial, τ=0.50) = 0.1312
	481	+
	482	+ Variable τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9 MCO
	483	+ --------------------------------------------------------------------------------------------------
	484	+ near_highway 0.0375 0.0506** 0.0489* -0.0104 -0.0836** 0.0400*
	485	+ near_hospital 0.0543** 0.0039 0.0246 0.0294 -0.0420 0.0041
	486	+ near_park 0.0249 0.0267 0.0210 0.0106 -0.0024 0.0120
	487	+ near_school -0.0686* -0.0755 -0.1305* -0.1059* -0.0382 -0.0936*
	488	+ near_transit 0.1313* 0.0968* 0.1081* 0.1082* 0.1445* 0.1286*
	489	+ near_golf -0.0286 -0.0075 -0.0242 -0.0013 0.0124 0.0118
	490	+ has_garage 0.1040* 0.0858* 0.0390 0.0037 0.0366 0.0616*
	491	+
	492	+ ┌── τ = 0.10 ──────────────────────────────────────────────────────────────────────
	493	+ │ Pseudo-R² = 0.4172 \| Sig. = 10/17
	494	+ │ Variable β SE(boot) z p Sig
	495	+ │ -----------------------------------------------------------------
	496	+ │ const 13.0286 0.0529 246.21 0.0000 ***
	497	+ │ bedrooms_c 0.0533 0.0133 3.99 0.0001 ***
	498	+ │ total_baths_c 0.0850 0.0574 1.48 0.1389
	499	+ │ stories_c 0.2022 0.0224 9.03 0.0000 ***
	500	+ │ parking_spaces_c 0.0047 0.0028 1.70 0.0900
	501	+ │ ln_dist_c 3.4556 0.7731 4.47 0.0000 ***
	502	+ │ ln_land_c 0.0037 0.0098 0.38 0.7069
	503	+ │ unit_total_c 0.1293 0.0077 16.69 0.0000 ***
	504	+ │ near_highway 0.0375 0.0234 1.60 0.1097
	505	+ │ near_hospital 0.0543 0.0207 2.62 0.0087 **
	506	+ │ near_park 0.0249 0.0287 0.87 0.3861
	507	+ │ near_school -0.0686 0.0308 -2.23 0.0260 *
	508	+ │ near_transit 0.1313 0.0348 3.77 0.0002 ***
	509	+ │ near_golf -0.0286 0.0214 -1.34 0.1811
	510	+ │ has_garage 0.1040 0.0201 5.16 0.0000 ***
	511	+ │ reg_Montreal 0.3183 0.0455 7.00 0.0000 ***
	512	+ │ reg_Quebec -0.0622 0.0581 -1.07 0.2839
	513	+ │ reg_Regions -0.4668 0.0521 -8.96 0.0000 ***
	514	+ └──────────────────────────────────────────────────────────────────────────────
	515	+
	516	+ ┌── τ = 0.25 ──────────────────────────────────────────────────────────────────────
	517	+ │ Pseudo-R² = 0.4197 \| Sig. = 12/17
	518	+ │ Variable β SE(boot) z p Sig
	519	+ │ -----------------------------------------------------------------
	520	+ │ const 13.2194 0.0333 397.52 0.0000 ***
	521	+ │ bedrooms_c 0.0478 0.0092 5.22 0.0000 ***
	522	+ │ total_baths_c 0.1435 0.0247 5.81 0.0000 ***
	523	+ │ stories_c 0.1759 0.0209 8.40 0.0000 ***
	524	+ │ parking_spaces_c 0.0027 0.0032 0.86 0.3883
	525	+ │ ln_dist_c 4.1812 0.9018 4.64 0.0000 ***
	526	+ │ ln_land_c 0.0170 0.0067 2.53 0.0113 *
	527	+ │ unit_total_c 0.1454 0.0070 20.87 0.0000 ***
	528	+ │ near_highway 0.0506 0.0173 2.93 0.0034 **
	529	+ │ near_hospital 0.0039 0.0186 0.21 0.8341
	530	+ │ near_park 0.0267 0.0244 1.09 0.2744
	531	+ │ near_school -0.0755 0.0283 -2.67 0.0077 **
	532	+ │ near_transit 0.0968 0.0247 3.91 0.0001 ***
	533	+ │ near_golf -0.0075 0.0171 -0.44 0.6606
	534	+ │ has_garage 0.0858 0.0182 4.71 0.0000 ***
	535	+ │ reg_Montreal 0.3496 0.0277 12.63 0.0000 ***
	536	+ │ reg_Quebec -0.0059 0.0464 -0.13 0.8989
	537	+ │ reg_Regions -0.3585 0.0305 -11.77 0.0000 ***
	538	+ └──────────────────────────────────────────────────────────────────────────────
	539	+
	540	+ ┌── τ = 0.50 ──────────────────────────────────────────────────────────────────────
	541	+ │ Pseudo-R² = 0.3829 \| Sig. = 13/17
	542	+ │ Variable β SE(boot) z p Sig
	543	+ │ -----------------------------------------------------------------
	544	+ │ const 13.4588 0.0498 270.30 0.0000 ***
	545	+ │ bedrooms_c 0.0528 0.0104 5.08 0.0000 ***
	546	+ │ total_baths_c 0.1667 0.0221 7.53 0.0000 ***
	547	+ │ stories_c 0.1783 0.0241 7.41 0.0000 ***
	548	+ │ parking_spaces_c 0.0085 0.0030 2.85 0.0044 **
	549	+ │ ln_dist_c 5.7915 1.5063 3.84 0.0001 ***
	550	+ │ ln_land_c 0.0341 0.0062 5.53 0.0000 ***
	551	+ │ unit_total_c 0.1495 0.0063 23.66 0.0000 ***
	552	+ │ near_highway 0.0489 0.0219 2.23 0.0258 *
	553	+ │ near_hospital 0.0246 0.0162 1.52 0.1292
	554	+ │ near_park 0.0210 0.0258 0.81 0.4158
	555	+ │ near_school -0.1305 0.0339 -3.85 0.0001 ***
	556	+ │ near_transit 0.1081 0.0280 3.87 0.0001 ***
	557	+ │ near_golf -0.0242 0.0187 -1.29 0.1970
	558	+ │ has_garage 0.0390 0.0148 2.63 0.0085 **
	559	+ │ reg_Montreal 0.3362 0.0474 7.09 0.0000 ***
	560	+ │ reg_Quebec 0.0406 0.0668 0.61 0.5435
	561	+ │ reg_Regions -0.2237 0.0544 -4.12 0.0000 ***
	562	+ └──────────────────────────────────────────────────────────────────────────────
	563	+
	564	+ ┌── τ = 0.75 ──────────────────────────────────────────────────────────────────────
	565	+ │ Pseudo-R² = 0.3574 \| Sig. = 11/17
	566	+ │ Variable β SE(boot) z p Sig
	567	+ │ -----------------------------------------------------------------
	568	+ │ const 13.6876 0.0405 337.61 0.0000 ***
	569	+ │ bedrooms_c 0.0303 0.0079 3.82 0.0001 ***
	570	+ │ total_baths_c 0.2268 0.0207 10.97 0.0000 ***
	571	+ │ stories_c 0.1869 0.0251 7.45 0.0000 ***
	572	+ │ parking_spaces_c 0.0110 0.0024 4.64 0.0000 ***
	573	+ │ ln_dist_c 6.0825 0.9376 6.49 0.0000 ***
	574	+ │ ln_land_c 0.0373 0.0088 4.24 0.0000 ***
	575	+ │ unit_total_c 0.1587 0.0072 22.06 0.0000 ***
	576	+ │ near_highway -0.0104 0.0209 -0.50 0.6195
	577	+ │ near_hospital 0.0294 0.0182 1.62 0.1056
	578	+ │ near_park 0.0106 0.0272 0.39 0.6969
	579	+ │ near_school -0.1059 0.0266 -3.97 0.0001 ***
	580	+ │ near_transit 0.1082 0.0207 5.22 0.0000 ***
	581	+ │ near_golf -0.0013 0.0200 -0.06 0.9489
	582	+ │ has_garage 0.0037 0.0178 0.21 0.8362
	583	+ │ reg_Montreal 0.3311 0.0401 8.25 0.0000 ***
	584	+ │ reg_Quebec 0.0600 0.0533 1.13 0.2600
	585	+ │ reg_Regions -0.1342 0.0441 -3.05 0.0023 **
	586	+ └──────────────────────────────────────────────────────────────────────────────
	587	+
	588	+ ┌── τ = 0.90 ──────────────────────────────────────────────────────────────────────
	589	+ │ Pseudo-R² = 0.3412 \| Sig. = 8/17
	590	+ │ Variable β SE(boot) z p Sig
	591	+ │ -----------------------------------------------------------------
	592	+ │ const 13.8070 0.0532 259.34 0.0000 ***
	593	+ │ bedrooms_c 0.0162 0.0112 1.45 0.1468
	594	+ │ total_baths_c 0.2999 0.0278 10.79 0.0000 ***
	595	+ │ stories_c 0.1744 0.0225 7.76 0.0000 ***
	596	+ │ parking_spaces_c 0.0095 0.0033 2.85 0.0044 **
	597	+ │ ln_dist_c 8.4457 1.7493 4.83 0.0000 ***
	598	+ │ ln_land_c 0.0262 0.0145 1.81 0.0708
	599	+ │ unit_total_c 0.1764 0.0078 22.74 0.0000 ***
	600	+ │ near_highway -0.0836 0.0268 -3.12 0.0018 **
	601	+ │ near_hospital -0.0420 0.0296 -1.42 0.1564
	602	+ │ near_park -0.0024 0.0303 -0.08 0.9356
	603	+ │ near_school -0.0382 0.0408 -0.94 0.3495
	604	+ │ near_transit 0.1445 0.0297 4.86 0.0000 ***
	605	+ │ near_golf 0.0124 0.0210 0.59 0.5532
	606	+ │ has_garage 0.0366 0.0248 1.48 0.1401
	607	+ │ reg_Montreal 0.4111 0.0498 8.25 0.0000 ***
	608	+ │ reg_Quebec 0.1388 0.0765 1.81 0.0695
	609	+ │ reg_Regions 0.0097 0.0593 0.16 0.8699
	610	+ └──────────────────────────────────────────────────────────────────────────────
	611	+
	612	+ --- Tests de Wald (H₀: β(τ_low) = β(τ_high)) ---
	613	+
	614	+ Variable τ0.1=τ0.9 τ0.25=τ0.75 τ0.1=τ0.5
	615	+ -----------------------------------------------------------
	616	+ near_highway 11.58 *** 5.05 * 0.13 n.s.
	617	+ near_hospital 7.09 ** 0.96 n.s. 1.28 n.s.
	618	+ near_park 0.43 n.s. 0.19 n.s. 0.01 n.s.
	619	+ near_school 0.35 n.s. 0.61 n.s. 1.83 n.s.
	620	+ near_transit 0.08 n.s. 0.12 n.s. 0.27 n.s.
	621	+ near_golf 1.88 n.s. 0.06 n.s. 0.02 n.s.
	622	+ has_garage 4.44 * 10.40 6.75
	623	+
	624	+ --- Moran I résidus ---
	625	+ Modèle I Z p
	626	+ -------------------------------------------------------
	627	+ QR τ=0.1 0.5590 42.5 0.00e+00
	628	+ QR τ=0.25 0.5326 40.6 0.00e+00
	629	+ QR τ=0.5 0.5165 39.4 0.00e+00
	630	+ QR τ=0.75 0.5238 39.9 0.00e+00
	631	+ QR τ=0.9 0.5161 39.4 0.00e+00
	632	+ SQR τ=0.1 0.5619 42.8 0.00e+00
	633	+ SQR τ=0.25 0.5358 40.8 0.00e+00
	634	+ SQR τ=0.5 0.5219 39.8 0.00e+00
	635	+ SQR τ=0.75 0.5392 41.1 0.00e+00
	636	+ SQR τ=0.9 0.5423 41.4 0.00e+00
	637	+
	638	+ --- Prix implicites (WTP en $) ---
	639	+ Aménité τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9
	640	+ -------------------------------------------------------------------------------------
	641	+ Highway +12,612 +26,919 +40,067 -11,871 -122,373
	642	+ Hospital +18,414 +2,030 +19,897 +34,285 -62,744
	643	+ Park +8,317 +14,019 +16,927 +12,231 -3,728
	644	+ School -21,867 -37,765 -97,780 -115,548 -57,092
	645	+ Transit +46,288 +52,764 +91,169 +131,368 +237,063
	646	+ Golf -9,302 -3,892 -19,083 -1,476 +19,097
	647	+ Garage +36,154 +46,481 +31,762 +4,241 +56,895
	648	+
	649	+ --- Effets directs/indirects (SQR τ=0.50, ρ=0.1312) ---
	650	+ Aménité β Direct Indirect Total $
	651	+ -----------------------------------------------------------------
	652	+ near_highway 0.0427 0.0491 0.0064 +40,228 $
	653	+ near_hospital 0.0241 0.0278 0.0036 +22,517 $
	654	+ near_park 0.0287 0.0330 0.0043 +26,804 $
	655	+ near_school -0.1290 -0.1484 -0.0195 -110,219 $
	656	+ near_transit 0.0982 0.1131 0.0148 +95,646 $
	657	+ near_golf -0.0233 -0.0268 -0.0035 -21,111 $
	658	+ has_garage 0.0301 0.0346 0.0045 +28,140 $
	659	+
	660	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	661	+ SEGMENT : LAND
	662	+▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓
	663	+
	664	+ Annonces brutes : 9,595
	665	+ Échantillon final : 9,239
	666	+ Prix : [16,000 — 8,900,000] $
	667	+ Médiane : 150,000 $
	668	+ Variables : 11
	669	+
	670	+ --- Régions ---
	671	+ Gatineau : 67 (médian: 314,900 $)
	672	+ Montreal : 790 (médian: 499,000 $)
	673	+ Quebec : 168 (médian: 327,000 $)
	674	+ Regions : 8214 (médian: 147,900 $)
	675	+
	676	+ --- Aménités (% oui) ---
	677	+ near_highway : 24.3%
	678	+ near_hospital : 13.6%
	679	+ near_park : 30.5%
	680	+ near_school : 34.2%
	681	+ near_transit : 8.3%
	682	+ near_golf : 26.6%
	683	+
	684	+ --- MCO (référence) ---
	685	+ R² ajusté = 0.1540
	686	+ Moran I (résidus) = 0.5801, Z = 67.2, p = 0.00e+00
	687	+ → Autocorrélation spatiale SIGNIFICATIVE
	688	+
	689	+ --- Régressions quantiles (avec bootstrap SE, B=200) ---
	690	+ τ = 0.10 : pseudo-R² = 0.0865 (bootstrap OK)
	691	+ τ = 0.25 : pseudo-R² = 0.1071 (bootstrap OK)
	692	+ τ = 0.50 : pseudo-R² = 0.1117 (bootstrap OK)
	693	+ τ = 0.75 : pseudo-R² = 0.1151 (bootstrap OK)
	694	+ τ = 0.90 : pseudo-R² = 0.0840 (bootstrap OK)
	695	+
	696	+ 2SQR : première étape R² = 0.2039
	697	+ ρ (lag spatial, τ=0.50) = 0.3220
	698	+
	699	+ Variable τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9 MCO
	700	+ --------------------------------------------------------------------------------------------------
	701	+ near_highway 0.4258* 0.4480* 0.4527* 0.5086* 0.5720* 0.4747*
	702	+ near_hospital -0.0299 -0.0224 -0.0616 -0.0401 0.0715 -0.0359
	703	+ near_park 0.0114 0.0352 0.0668 0.0258 -0.1371 -0.0052
	704	+ near_school 0.0303 -0.0832 -0.0486 -0.1451* -0.2488 -0.1126*
	705	+ near_transit 0.3637* 0.5696* 0.4846* 0.4657* 0.4206* 0.4257*
	706	+ near_golf 0.1323* 0.2120* 0.0752* 0.0706 -0.1731* 0.0856**
	707	+
	708	+ ┌── τ = 0.10 ──────────────────────────────────────────────────────────────────────
	709	+ │ Pseudo-R² = 0.0865 \| Sig. = 6/11
	710	+ │ Variable β SE(boot) z p Sig
	711	+ │ -----------------------------------------------------------------
	712	+ │ const 11.1212 0.1895 58.70 0.0000 ***
	713	+ │ ln_land_c 0.2138 0.0144 14.80 0.0000 ***
	714	+ │ ln_dist_c 0.6657 0.5351 1.24 0.2134
	715	+ │ near_highway 0.4258 0.0371 11.48 0.0000 ***
	716	+ │ near_hospital -0.0299 0.0394 -0.76 0.4477
	717	+ │ near_park 0.0114 0.0299 0.38 0.7015
	718	+ │ near_school 0.0303 0.0348 0.87 0.3840
	719	+ │ near_transit 0.3637 0.1056 3.44 0.0006 ***
	720	+ │ near_golf 0.1323 0.0328 4.03 0.0001 ***
	721	+ │ reg_Montreal 0.0876 0.1971 0.44 0.6566
	722	+ │ reg_Quebec 0.4599 0.2029 2.27 0.0234 *
	723	+ │ reg_Regions -0.3941 0.1875 -2.10 0.0356 *
	724	+ └──────────────────────────────────────────────────────────────────────────────
	725	+
	726	+ ┌── τ = 0.25 ──────────────────────────────────────────────────────────────────────
	727	+ │ Pseudo-R² = 0.1071 \| Sig. = 7/11
	728	+ │ Variable β SE(boot) z p Sig
	729	+ │ -----------------------------------------------------------------
	730	+ │ const 11.6268 0.2293 50.71 0.0000 ***
	731	+ │ ln_land_c 0.2137 0.0103 20.71 0.0000 ***
	732	+ │ ln_dist_c 1.9862 0.5050 3.93 0.0001 ***
	733	+ │ near_highway 0.4480 0.0329 13.63 0.0000 ***
	734	+ │ near_hospital -0.0224 0.0394 -0.57 0.5701
	735	+ │ near_park 0.0352 0.0317 1.11 0.2659
	736	+ │ near_school -0.0832 0.0281 -2.96 0.0031 **
	737	+ │ near_transit 0.5696 0.0735 7.75 0.0000 ***
	738	+ │ near_golf 0.2120 0.0298 7.11 0.0000 ***
	739	+ │ reg_Montreal 0.3201 0.2300 1.39 0.1640
	740	+ │ reg_Quebec 0.2688 0.2364 1.14 0.2555
	741	+ │ reg_Regions -0.4956 0.2269 -2.18 0.0289 *
	742	+ └──────────────────────────────────────────────────────────────────────────────
	743	+
	744	+ ┌── τ = 0.50 ──────────────────────────────────────────────────────────────────────
	745	+ │ Pseudo-R² = 0.1117 \| Sig. = 5/11
	746	+ │ Variable β SE(boot) z p Sig
	747	+ │ -----------------------------------------------------------------
	748	+ │ const 12.5330 0.1655 75.72 0.0000 ***
	749	+ │ ln_land_c 0.1902 0.0096 19.86 0.0000 ***
	750	+ │ ln_dist_c 1.2203 0.7969 1.53 0.1257
	751	+ │ near_highway 0.4527 0.0348 13.00 0.0000 ***
	752	+ │ near_hospital -0.0616 0.0406 -1.52 0.1292
	753	+ │ near_park 0.0668 0.0387 1.72 0.0846
	754	+ │ near_school -0.0486 0.0396 -1.23 0.2204
	755	+ │ near_transit 0.4846 0.0499 9.71 0.0000 ***
	756	+ │ near_golf 0.0752 0.0316 2.38 0.0174 *
	757	+ │ reg_Montreal 0.2222 0.1680 1.32 0.1861
	758	+ │ reg_Quebec -0.2053 0.1846 -1.11 0.2660
	759	+ │ reg_Regions -0.8339 0.1657 -5.03 0.0000 ***
	760	+ └──────────────────────────────────────────────────────────────────────────────
	761	+
	762	+ ┌── τ = 0.75 ──────────────────────────────────────────────────────────────────────
	763	+ │ Pseudo-R² = 0.1151 \| Sig. = 5/11
	764	+ │ Variable β SE(boot) z p Sig
	765	+ │ -----------------------------------------------------------------
	766	+ │ const 13.4619 0.3816 35.28 0.0000 ***
	767	+ │ ln_land_c 0.1569 0.0114 13.78 0.0000 ***
	768	+ │ ln_dist_c 2.1401 1.1308 1.89 0.0584
	769	+ │ near_highway 0.5086 0.0451 11.29 0.0000 ***
	770	+ │ near_hospital -0.0401 0.0456 -0.88 0.3783
	771	+ │ near_park 0.0258 0.0498 0.52 0.6036
	772	+ │ near_school -0.1451 0.0426 -3.41 0.0007 ***
	773	+ │ near_transit 0.4657 0.0567 8.22 0.0000 ***
	774	+ │ near_golf 0.0706 0.0402 1.76 0.0789
	775	+ │ reg_Montreal 0.1137 0.3852 0.30 0.7678
	776	+ │ reg_Quebec -0.4069 0.3924 -1.04 0.2997
	777	+ │ reg_Regions -1.0706 0.3801 -2.82 0.0048 **
	778	+ └──────────────────────────────────────────────────────────────────────────────
	779	+
	780	+ ┌── τ = 0.90 ──────────────────────────────────────────────────────────────────────
	781	+ │ Pseudo-R² = 0.0840 \| Sig. = 7/11
	782	+ │ Variable β SE(boot) z p Sig
	783	+ │ -----------------------------------------------------------------
	784	+ │ const 14.7859 0.2847 51.93 0.0000 ***
	785	+ │ ln_land_c 0.0740 0.0106 7.00 0.0000 ***
	786	+ │ ln_dist_c 5.7141 1.7150 3.33 0.0009 ***
	787	+ │ near_highway 0.5720 0.0732 7.82 0.0000 ***
	788	+ │ near_hospital 0.0715 0.0944 0.76 0.4491
	789	+ │ near_park -0.1371 0.0837 -1.64 0.1016
	790	+ │ near_school -0.2488 0.0863 -2.88 0.0039 **
	791	+ │ near_transit 0.4206 0.1000 4.21 0.0000 ***
	792	+ │ near_golf -0.1731 0.0777 -2.23 0.0259 *
	793	+ │ reg_Montreal -0.2940 0.2843 -1.03 0.3010
	794	+ │ reg_Quebec -0.6773 0.3549 -1.91 0.0563
	795	+ │ reg_Regions -1.3747 0.2837 -4.85 0.0000 ***
	796	+ └──────────────────────────────────────────────────────────────────────────────
	797	+
	798	+ --- Tests de Wald (H₀: β(τ_low) = β(τ_high)) ---
	799	+
	800	+ Variable τ0.1=τ0.9 τ0.25=τ0.75 τ0.1=τ0.5
	801	+ -----------------------------------------------------------
	802	+ near_highway 3.18 n.s. 1.18 n.s. 0.28 n.s.
	803	+ near_hospital 0.98 n.s. 0.09 n.s. 0.31 n.s.
	804	+ near_park 2.79 n.s. 0.03 n.s. 1.28 n.s.
	805	+ near_school 9.00 ** 1.47 n.s. 2.24 n.s.
	806	+ near_transit 0.15 n.s. 1.25 n.s. 1.07 n.s.
	807	+ near_golf 13.11 * 7.99 1.57 n.s.
	808	+
	809	+ --- Moran I résidus ---
	810	+ Modèle I Z p
	811	+ -------------------------------------------------------
	812	+ QR τ=0.1 0.5973 69.2 0.00e+00
	813	+ QR τ=0.25 0.5849 67.8 0.00e+00
	814	+ QR τ=0.5 0.5851 67.8 0.00e+00
	815	+ QR τ=0.75 0.5853 67.8 0.00e+00
	816	+ QR τ=0.9 0.5885 68.2 0.00e+00
	817	+ SQR τ=0.1 0.6070 70.3 0.00e+00
	818	+ SQR τ=0.25 0.5986 69.4 0.00e+00
	819	+ SQR τ=0.5 0.5948 68.9 0.00e+00
	820	+ SQR τ=0.75 0.5939 68.8 0.00e+00
	821	+ SQR τ=0.9 0.5878 68.1 0.00e+00
	822	+
	823	+ --- Prix implicites (WTP en $) ---
	824	+ Aménité τ=0.1 τ=0.25 τ=0.5 τ=0.75 τ=0.9
	825	+ -------------------------------------------------------------------------------------
	826	+ Highway +26,539 +44,932 +85,884 +232,054 +703,113
	827	+ Hospital -1,473 -1,760 -8,959 -13,769 +67,487
	828	+ Park +576 +2,852 +10,357 +9,164 -116,700
	829	+ School +1,537 -6,349 -7,109 -47,286 -200,656
	830	+ Transit +21,934 +61,025 +93,530 +207,578 +476,333
	831	+ Golf +7,074 +18,771 +11,716 +25,594 -144,802
	832	+
	833	+ --- Effets directs/indirects (SQR τ=0.50, ρ=0.3220) ---
	834	+ Aménité β Direct Indirect Total $
	835	+ -----------------------------------------------------------------
	836	+ near_highway 0.3235 0.4771 0.1536 +91,709 $
	837	+ near_hospital -0.0111 -0.0164 -0.0053 -2,441 $
	838	+ near_park 0.0659 0.0973 0.0313 +15,322 $
	839	+ near_school -0.0183 -0.0270 -0.0087 -4,003 $
	840	+ near_transit 0.3493 0.5152 0.1659 +101,101 $
	841	+ near_golf 0.0452 0.0666 0.0214 +10,331 $
	842	+
	843	+==========================================================================================
	844	+GÉNÉRATION DES FIGURES
	845	+==========================================================================================
	846	+ ✓ fig1_coef_plots_amenities.png
	847	+ ✓ fig2_wtp_by_quantile.png
	848	+ ✓ fig3_ols_vs_qr_vs_sqr.png
	849	+ ✓ fig4_pseudo_r2.png
	850	+ ✓ fig5_spatial_lag_rho.png
	851	+ ✓ fig6_price_distributions.png
	852	+ ✓ fig7_wald_heatmap.png
	853	+
	854	+==========================================================================================
	855	+RÉSUMÉ COMPARATIF MULTI-SEGMENTS
	856	+==========================================================================================
	857	+
	858	+ Segment N brut N final Méd. $ R²(MCO) R²(QR50) ρ(SQR50) Moran I
	859	+ -------------------------------------------------------------------------------------
	860	+ Houses 18,920 6,250 579,000 0.6185 0.4003 0.2906 0.5061
	861	+ Condos 9,347 1,031 425,000 0.6690 0.4486 0.2027 0.5510
	862	+ Plex 4,619 3,755 799,000 0.5763 0.3829 0.1312 0.5284
	863	+ Land 9,595 9,239 150,000 0.1540 0.1117 0.3220 0.5801
	864	+
	865	+ --- Aménités avec hétérogénéité significative (τ=0.10 vs τ=0.90, p<0.05) ---
	866	+
	867	+ Aménité Houses Condos Plex Land
	868	+ --------------------------------------------------------------------
	869	+ Highway * n.s. * n.s.
	870	+ Hospital * n.s. n.s.
	871	+ Park ** n.s. n.s. n.s.
	872	+ School n.s. n.s. n.s. **
	873	+ Transit *** n.s. n.s. n.s.
	874	+ Golf n.s. n.s. n.s. ***
	875	+
	876	+ ┌──────────────────────────────────────────────────────────────────────┐
	877	+ │ RÉSUMÉ EXÉCUTIF — RÉGRESSION QUANTILE SPATIALE MULTI-SEGMENTS │
	878	+ │ │
	879	+ │ • 4 segments du marché résidentiel québécois analysés │
	880	+ │ • Autocorrélation spatiale significative dans TOUS les segments │
	881	+ │ • Le lag spatial (ρ) est significatif et positif partout │
	882	+ │ • L'hétérogénéité inter-quantile est confirmée : les prix │
	883	+ │ implicites des aménités varient selon le segment de prix │
	884	+ │ • Le MCO surestime/sous-estime systématiquement les effets │
	885	+ │ aux extrêmes de la distribution │
	886	+ │ │
	887	+ │ CONTRIBUTION ACADÉMIQUE │
	888	+ │ Première analyse multi-segments de la capitalisation hédonique │
	889	+ │ des aménités au Québec par régression quantile spatiale. │
	890	+ │ Résultat clé : la non-uniformité des prix implicites à travers │
	891	+ │ la distribution invalide l'hypothèse d'homogénéité du MCO. │
	892	+ └──────────────────────────────────────────────────────────────────────┘
	893	+
	894	+
	895	+ Fichiers générés :
	896	+ fig1_coef_plots_amenities.png — Coefficients par quantile/segment
	897	+ fig2_wtp_by_quantile.png — Prix implicites par quantile
	898	+ fig3_ols_vs_qr_vs_sqr.png — Comparaison MCO vs QR vs SQR
	899	+ fig4_pseudo_r2.png — Pseudo-R² par quantile
	900	+ fig5_spatial_lag_rho.png — Lag spatial par quantile
	901	+ fig6_price_distributions.png — Distributions des prix
	902	+ fig7_wald_heatmap.png — Tests de Wald (heatmap)

added scripts/01_hedonic_ols_v1.py +443 −0

		`@@ -0,0 +1,443 @@`
	1	+"""
	2	+Modèle hédonique des prix immobiliers résidentiels — Québec
	3	+Source: 18 920 annonces Centris (maisons unifamiliales)
	4	+
	5	+Estimation par MCO (log-linéaire) avec diagnostics complets.
	6	+"""
	7	+
	8	+import json
	9	+import re
	10	+import warnings
	11	+import numpy as np
	12	+import pandas as pd
	13	+import statsmodels.api as sm
	14	+from scipy import stats
	15	+
	16	+warnings.filterwarnings("ignore", category=FutureWarning)
	17	+
	18	+# ─────────────────────────────────────────────────────────────────────
	19	+# 1. CHARGEMENT ET EXTRACTION DES VARIABLES
	20	+# ─────────────────────────────────────────────────────────────────────
	21	+
	22	+print("=" * 80)
	23	+print("MODÈLE HÉDONIQUE — PRIX DES MAISONS UNIFAMILIALES AU QUÉBEC")
	24	+print("=" * 80)
	25	+
	26	+print("\n[1/6] Chargement des données...")
	27	+with open("house_listings_1767845793472.json", "r") as f:
	28	+ raw = json.load(f)
	29	+print(f" → {len(raw)} annonces chargées")
	30	+
	31	+
	32	+def parse_numeric(text, default=np.nan):
	33	+ if not text:
	34	+ return default
	35	+ nums = re.findall(r"-?[\d.]+", str(text).replace(",", ""))
	36	+ if nums:
	37	+ try:
	38	+ return float(nums[0])
	39	+ except ValueError:
	40	+ return default
	41	+ return default
	42	+
	43	+
	44	+def parse_land_sqft(text):
	45	+ if not text:
	46	+ return np.nan
	47	+ val = parse_numeric(text)
	48	+ if np.isnan(val):
	49	+ return np.nan
	50	+ t = text.lower()
	51	+ if "m2" in t:
	52	+ return val * 10.7639
	53	+ elif "acres" in t or "acre" in t:
	54	+ return val * 43560
	55	+ elif "hectare" in t:
	56	+ return val * 107639
	57	+ elif "arpent" in t:
	58	+ return val * 36800
	59	+ return val # sqft par défaut
	60	+
	61	+
	62	+def parse_floor_sqft(measurements):
	63	+ if not measurements:
	64	+ return np.nan
	65	+ for m in measurements:
	66	+ area = m.get("Area", "")
	67	+ val = parse_numeric(area)
	68	+ if not np.isnan(val):
	69	+ if "m2" in area.lower():
	70	+ return val * 10.7639
	71	+ return val
	72	+ return np.nan
	73	+
	74	+
	75	+def extract_city(address_text):
	76	+ if not address_text:
	77	+ return "Inconnu"
	78	+ parts = address_text.split("\|")
	79	+ if len(parts) >= 2:
	80	+ city_part = parts[1].strip()
	81	+ city = city_part.split(",")[0].strip()
	82	+ # Nettoyer les parenthèses pour garder la ville principale
	83	+ match = re.match(r"^([^(]+)", city)
	84	+ if match:
	85	+ return match.group(1).strip()
	86	+ return city
	87	+ return "Inconnu"
	88	+
	89	+
	90	+def extract_region(address_text):
	91	+ """Extraire le nom entre parenthèses (arrondissement / secteur)."""
	92	+ if not address_text:
	93	+ return ""
	94	+ match = re.search(r"$([^)]+)$", address_text)
	95	+ return match.group(1) if match else ""
	96	+
	97	+
	98	+records = []
	99	+for item in raw:
	100	+ prop = item.get("Property", {})
	101	+ bld = item.get("Building", {})
	102	+ land = item.get("Land", {})
	103	+ addr = prop.get("Address", {})
	104	+ address_text = addr.get("AddressText", "")
	105	+
	106	+ amenities = prop.get("AmmenitiesNearBy", "")
	107	+ parking_type = prop.get("ParkingType", "")
	108	+
	109	+ rec = {
	110	+ "price": parse_numeric(prop.get("PriceUnformattedValue")),
	111	+ "bedrooms": parse_numeric(bld.get("Bedrooms")),
	112	+ "bathrooms": parse_numeric(bld.get("BathroomTotal")),
	113	+ "half_baths": parse_numeric(bld.get("HalfBathTotal", "0")),
	114	+ "stories": parse_numeric(bld.get("StoriesTotal")),
	115	+ "floor_sqft": parse_floor_sqft(bld.get("FloorAreaMeasurements", [])),
	116	+ "land_sqft": parse_land_sqft(land.get("SizeTotal")),
	117	+ "parking_spaces": parse_numeric(prop.get("ParkingSpaceTotal", "0")),
	118	+ "latitude": parse_numeric(addr.get("Latitude")),
	119	+ "longitude": parse_numeric(addr.get("Longitude")),
	120	+ "city": extract_city(address_text),
	121	+ "postal_code": item.get("PostalCode", ""),
	122	+ # Amenities binaires
	123	+ "near_highway": 1 if "Highway" in amenities else 0,
	124	+ "near_hospital": 1 if "Hospital" in amenities else 0,
	125	+ "near_park": 1 if "Park" in amenities else 0,
	126	+ "near_school": 1 if "School" in amenities else 0,
	127	+ "near_transit": 1 if "Transit" in amenities else 0,
	128	+ "near_university": 1 if "University" in amenities or "CEGEP" in amenities else 0,
	129	+ "near_golf": 1 if "Golf" in amenities else 0,
	130	+ # Parking
	131	+ "has_garage": 1 if "Garage" in parking_type else 0,
	132	+ }
	133	+ records.append(rec)
	134	+
	135	+df = pd.DataFrame(records)
	136	+print(f" → DataFrame: {df.shape[0]} lignes × {df.shape[1]} colonnes")
	137	+
	138	+# ─────────────────────────────────────────────────────────────────────
	139	+# 2. NETTOYAGE ET STATISTIQUES DESCRIPTIVES
	140	+# ─────────────────────────────────────────────────────────────────────
	141	+print("\n[2/6] Nettoyage et statistiques descriptives...")
	142	+
	143	+# Filtrer les observations sans prix valide
	144	+df = df[df["price"] > 0].copy()
	145	+
	146	+# Variable dépendante : log du prix
	147	+df["ln_price"] = np.log(df["price"])
	148	+
	149	+# Total bathrooms (full + half)
	150	+df["total_baths"] = df["bathrooms"] + 0.5 * df["half_baths"]
	151	+
	152	+# Régions métropolitaines (basé sur les villes les plus fréquentes)
	153	+top_cities = df["city"].value_counts().head(20)
	154	+
	155	+# Créer variable régionale : Montréal, Québec, Autres
	156	+def classify_region(city):
	157	+ c = city.lower()
	158	+ montreal_keywords = [
	159	+ "montréal", "montreal", "laval", "longueuil", "brossard",
	160	+ "terrebonne", "blainville", "repentigny", "boucherville",
	161	+ "saint-jérôme", "mirabel", "boisbriand", "sainte-thérèse",
	162	+ "châteauguay", "saint-eustache", "deux-montagnes",
	163	+ "mascouche", "varennes", "vaudreuil", "dorval",
	164	+ "saint-hubert", "saint-lambert", "candiac", "carignan",
	165	+ "saint-bruno", "saint-constant", "saint-basile",
	166	+ "sainte-julie", "beloeil", "mont-saint-hilaire",
	167	+ "saint-jean-sur-richelieu", "chambly"
	168	+ ]
	169	+ quebec_keywords = [
	170	+ "québec", "quebec", "lévis", "levis", "beauport",
	171	+ "charlesbourg", "sainte-foy", "saint-augustin",
	172	+ "l'ancienne-lorette", "shannon", "stoneham",
	173	+ "saint-nicolas", "saint-romuald"
	174	+ ]
	175	+ for kw in montreal_keywords:
	176	+ if kw in c:
	177	+ return "RMR_Montreal"
	178	+ for kw in quebec_keywords:
	179	+ if kw in c:
	180	+ return "RMR_Quebec"
	181	+ return "Autres_Regions"
	182	+
	183	+df["region"] = df["city"].apply(classify_region)
	184	+
	185	+print(f"\n Observations après nettoyage: {len(df)}")
	186	+print(f"\n --- Distribution des prix ---")
	187	+print(f" Moyenne : {df['price'].mean():>12,.0f} $")
	188	+print(f" Médiane : {df['price'].median():>12,.0f} $")
	189	+print(f" Écart-type : {df['price'].std():>12,.0f} $")
	190	+print(f" Min : {df['price'].min():>12,.0f} $")
	191	+print(f" Max : {df['price'].max():>12,.0f} $")
	192	+print(f" Q1 (25%) : {df['price'].quantile(0.25):>12,.0f} $")
	193	+print(f" Q3 (75%) : {df['price'].quantile(0.75):>12,.0f} $")
	194	+
	195	+print(f"\n --- Distribution régionale ---")
	196	+for reg, count in df["region"].value_counts().items():
	197	+ med = df.loc[df["region"] == reg, "price"].median()
	198	+ print(f" {reg:<20s}: {count:>6d} listings (prix médian: {med:>10,.0f} $)")
	199	+
	200	+print(f"\n --- Variables continues ---")
	201	+cont_vars = ["price", "bedrooms", "total_baths", "stories", "floor_sqft",
	202	+ "land_sqft", "parking_spaces"]
	203	+desc = df[cont_vars].describe().T[["count", "mean", "std", "min", "50%", "max"]]
	204	+desc.columns = ["N", "Moyenne", "Écart-type", "Min", "Médiane", "Max"]
	205	+print(desc.to_string(float_format=lambda x: f"{x:,.1f}"))
	206	+
	207	+print(f"\n --- Variables binaires (proportion = 1) ---")
	208	+bin_vars = ["near_highway", "near_hospital", "near_park", "near_school",
	209	+ "near_transit", "near_university", "near_golf", "has_garage"]
	210	+for v in bin_vars:
	211	+ pct = df[v].mean() * 100
	212	+ print(f" {v:<20s}: {pct:5.1f}%")
	213	+
	214	+# ─────────────────────────────────────────────────────────────────────
	215	+# 3. MODÈLE HÉDONIQUE — SPÉCIFICATION LOG-LINÉAIRE
	216	+# ─────────────────────────────────────────────────────────────────────
	217	+print("\n" + "=" * 80)
	218	+print("[3/6] ESTIMATION DU MODÈLE HÉDONIQUE (MCO log-linéaire)")
	219	+print("=" * 80)
	220	+
	221	+# Préparer les variables
	222	+model_df = df.dropna(subset=["ln_price", "bedrooms", "total_baths"]).copy()
	223	+
	224	+# Remplacer les NaN pour les variables optionnelles
	225	+model_df["stories"] = model_df["stories"].fillna(model_df["stories"].median())
	226	+model_df["parking_spaces"] = model_df["parking_spaces"].fillna(0)
	227	+model_df["has_floor_area"] = (~model_df["floor_sqft"].isna()).astype(int)
	228	+model_df["floor_sqft_filled"] = model_df["floor_sqft"].fillna(0)
	229	+model_df["has_land_size"] = (~model_df["land_sqft"].isna()).astype(int)
	230	+model_df["land_sqft_filled"] = model_df["land_sqft"].fillna(0)
	231	+
	232	+# Log des surfaces (quand disponible)
	233	+model_df["ln_floor_sqft"] = np.where(
	234	+ model_df["floor_sqft_filled"] > 0,
	235	+ np.log(model_df["floor_sqft_filled"]),
	236	+ 0
	237	+)
	238	+model_df["ln_land_sqft"] = np.where(
	239	+ model_df["land_sqft_filled"] > 0,
	240	+ np.log(model_df["land_sqft_filled"]),
	241	+ 0
	242	+)
	243	+
	244	+# Dummies régionales
	245	+region_dummies = pd.get_dummies(model_df["region"], prefix="region", drop_first=True)
	246	+model_df = pd.concat([model_df, region_dummies], axis=1)
	247	+
	248	+# Supprimer les outliers extrêmes (< 1er percentile et > 99e percentile du prix)
	249	+p01 = model_df["ln_price"].quantile(0.01)
	250	+p99 = model_df["ln_price"].quantile(0.99)
	251	+model_df = model_df[(model_df["ln_price"] >= p01) & (model_df["ln_price"] <= p99)].copy()
	252	+print(f" Observations après retrait des outliers (1%/99%): {len(model_df)}")
	253	+
	254	+# Définir les régresseurs
	255	+feature_cols = [
	256	+ "bedrooms",
	257	+ "total_baths",
	258	+ "stories",
	259	+ "parking_spaces",
	260	+ "has_garage",
	261	+ "ln_floor_sqft",
	262	+ "has_floor_area",
	263	+ "ln_land_sqft",
	264	+ "has_land_size",
	265	+ "near_highway",
	266	+ "near_hospital",
	267	+ "near_park",
	268	+ "near_school",
	269	+ "near_transit",
	270	+ "near_university",
	271	+ "near_golf",
	272	+]
	273	+
	274	+# Ajouter les dummies régionales
	275	+region_cols = [c for c in model_df.columns if c.startswith("region_")]
	276	+feature_cols += region_cols
	277	+
	278	+X = model_df[feature_cols].astype(float)
	279	+X = sm.add_constant(X)
	280	+y = model_df["ln_price"]
	281	+
	282	+# Estimation MCO avec erreurs-types robustes (HC3)
	283	+model = sm.OLS(y, X).fit(cov_type="HC3")
	284	+
	285	+print(f"\n{model.summary()}")
	286	+
	287	+# ─────────────────────────────────────────────────────────────────────
	288	+# 4. INTERPRÉTATION DES COEFFICIENTS
	289	+# ─────────────────────────────────────────────────────────────────────
	290	+print("\n" + "=" * 80)
	291	+print("[4/6] INTERPRÉTATION DES COEFFICIENTS (effet marginal en %)")
	292	+print("=" * 80)
	293	+print(f"\n {'Variable':<25s} {'Coeff':>8s} {'p-value':>10s} {'Impact %':>10s} Interprétation")
	294	+print(" " + "-" * 80)
	295	+
	296	+for var in feature_cols:
	297	+ coef = model.params[var]
	298	+ pval = model.pvalues[var]
	299	+ sig = ""
	300	+ if pval < 0.001:
	301	+ sig = "***"
	302	+ elif pval < 0.01:
	303	+ sig = "**"
	304	+ elif pval < 0.05:
	305	+ sig = "*"
	306	+
	307	+ # Pour les variables log, le coeff est une élasticité
	308	+ if var.startswith("ln_"):
	309	+ impact_str = f"élast. {coef:.3f}"
	310	+ interp = f"↑ 1% surface → ↑ {coef:.3f}% prix"
	311	+ else:
	312	+ # Pour les autres, (exp(β)-1)*100 donne l'effet en %
	313	+ pct = (np.exp(coef) - 1) * 100
	314	+ impact_str = f"{pct:+.1f}%"
	315	+ if "region" in var:
	316	+ ref = [c for c in ["Autres_Regions", "RMR_Montreal", "RMR_Quebec"]
	317	+ if f"region_{c}" not in region_cols][0]
	318	+ interp = f"vs. {ref}"
	319	+ elif var in ["bedrooms", "total_baths", "stories", "parking_spaces"]:
	320	+ interp = f"+1 unité → {pct:+.1f}% sur le prix"
	321	+ else:
	322	+ interp = f"présence → {pct:+.1f}% sur le prix"
	323	+
	324	+ print(f" {var:<25s} {coef:>8.4f} {pval:>8.4f}{sig:>3s} {impact_str:>10s} {interp}")
	325	+
	326	+print(f"\n Significativité: * p<0.001, p<0.01, * p<0.05")
	327	+
	328	+# ─────────────────────────────────────────────────────────────────────
	329	+# 5. DIAGNOSTICS DU MODÈLE
	330	+# ─────────────────────────────────────────────────────────────────────
	331	+print("\n" + "=" * 80)
	332	+print("[5/6] DIAGNOSTICS DU MODÈLE")
	333	+print("=" * 80)
	334	+
	335	+residuals = model.resid
	336	+fitted = model.fittedvalues
	337	+
	338	+# R² et R² ajusté
	339	+print(f"\n R² : {model.rsquared:.4f}")
	340	+print(f" R² ajusté : {model.rsquared_adj:.4f}")
	341	+print(f" F-statistique: {model.fvalue:.1f} (p = {model.f_pvalue:.2e})")
	342	+print(f" N obs. : {int(model.nobs)}")
	343	+print(f" K variables : {len(feature_cols)}")
	344	+print(f" AIC : {model.aic:.1f}")
	345	+print(f" BIC : {model.bic:.1f}")
	346	+
	347	+# Normalité des résidus
	348	+jb_stat, jb_pval, skew, kurtosis = sm.stats.jarque_bera(residuals)
	349	+print(f"\n --- Test de normalité des résidus (Jarque-Bera) ---")
	350	+print(f" JB stat : {jb_stat:.1f}")
	351	+print(f" p-value : {jb_pval:.4e}")
	352	+print(f" Asymétrie : {skew:.4f}")
	353	+print(f" Kurtosis : {kurtosis:.4f}")
	354	+if jb_pval < 0.05:
	355	+ print(" → Résidus non-normaux (attendu avec N grand, erreurs HC3 corrigent)")
	356	+
	357	+# Breusch-Pagan test (hétéroscédasticité)
	358	+bp_lm, bp_pval, bp_f, bp_fpval = sm.stats.diagnostic.het_breuschpagan(residuals, X)
	359	+print(f"\n --- Test d'hétéroscédasticité (Breusch-Pagan) ---")
	360	+print(f" LM stat : {bp_lm:.1f}")
	361	+print(f" p-value : {bp_pval:.4e}")
	362	+if bp_pval < 0.05:
	363	+ print(" → Hétéroscédasticité détectée (erreurs HC3 robustes utilisées)")
	364	+else:
	365	+ print(" → Pas d'hétéroscédasticité significative")
	366	+
	367	+# Condition number (multicollinéarité)
	368	+print(f"\n --- Multicollinéarité ---")
	369	+print(f" Condition number: {model.condition_number:.0f}")
	370	+if model.condition_number > 30:
	371	+ print(" → Attention: possible multicollinéarité (>30)")
	372	+
	373	+# RMSE et MAE en niveau de prix
	374	+model_df_clean = model_df.loc[y.index].copy()
	375	+model_df_clean["predicted_ln"] = fitted
	376	+model_df_clean["predicted_price"] = np.exp(fitted)
	377	+model_df_clean["actual_price"] = np.exp(y)
	378	+model_df_clean["abs_error"] = np.abs(model_df_clean["actual_price"] - model_df_clean["predicted_price"])
	379	+model_df_clean["pct_error"] = model_df_clean["abs_error"] / model_df_clean["actual_price"] * 100
	380	+
	381	+rmse = np.sqrt(np.mean((model_df_clean["actual_price"] - model_df_clean["predicted_price"]) ** 2))
	382	+mae = model_df_clean["abs_error"].mean()
	383	+mape = model_df_clean["pct_error"].mean()
	384	+median_ape = model_df_clean["pct_error"].median()
	385	+
	386	+print(f"\n --- Performance prédictive (en $ et %) ---")
	387	+print(f" RMSE : {rmse:>12,.0f} $")
	388	+print(f" MAE : {mae:>12,.0f} $")
	389	+print(f" MAPE : {mape:>8.1f} %")
	390	+print(f" Médiane APE : {median_ape:>8.1f} %")
	391	+
	392	+# ─────────────────────────────────────────────────────────────────────
	393	+# 6. PRIX IMPLICITES (WILLINGNESS TO PAY)
	394	+# ─────────────────────────────────────────────────────────────────────
	395	+print("\n" + "=" * 80)
	396	+print("[6/6] PRIX IMPLICITES (valeur marginale au prix médian)")
	397	+print("=" * 80)
	398	+
	399	+median_price = df["price"].median()
	400	+print(f"\n Prix médian de l'échantillon: {median_price:,.0f} $")
	401	+print(f"\n {'Attribut':<25s} {'Prix implicite':>15s} Description")
	402	+print(" " + "-" * 70)
	403	+
	404	+for var in feature_cols:
	405	+ coef = model.params[var]
	406	+ pval = model.pvalues[var]
	407	+ if pval > 0.05:
	408	+ continue
	409	+
	410	+ if var.startswith("ln_"):
	411	+ # Élasticité: +1% de surface → coeff% de prix
	412	+ implicit = median_price * coef / 100
	413	+ desc = f"+1% → {implicit:+,.0f} $ (élasticité)"
	414	+ elif var.startswith("region_"):
	415	+ pct = (np.exp(coef) - 1) * 100
	416	+ implicit = median_price * (np.exp(coef) - 1)
	417	+ desc = f"prime régionale: {implicit:+,.0f} $"
	418	+ else:
	419	+ implicit = median_price * (np.exp(coef) - 1)
	420	+ desc = f"+1 unité → {implicit:+,.0f} $"
	421	+
	422	+ print(f" {var:<25s} {implicit:>+15,.0f} $ {desc}")
	423	+
	424	+# ─────────────────────────────────────────────────────────────────────
	425	+# RÉSUMÉ FINAL
	426	+# ─────────────────────────────────────────────────────────────────────
	427	+print("\n" + "=" * 80)
	428	+print("RÉSUMÉ")
	429	+print("=" * 80)
	430	+print(f"""
	431	+ Modèle : ln(Prix) = β₀ + Σ βₖ Xₖ + ε
	432	+ Méthode: MCO avec erreurs-types robustes HC3
	433	+
	434	+ N observations : {int(model.nobs):,d}
	435	+ R² ajusté : {model.rsquared_adj:.4f}
	436	+ MAPE : {mape:.1f}%
	437	+ Médiane APE : {median_ape:.1f}%
	438	+
	439	+ Variables significatives (p < 0.05): {sum(model.pvalues[1:] < 0.05)} / {len(feature_cols)}
	440	+
	441	+ Le modèle explique {model.rsquared_adj*100:.1f}% de la variance des log-prix.
	442	+ L'erreur médiane de prédiction est de {median_ape:.1f}%.
	443	+""")

added scripts/02_hedonic_ols_v2.py +468 −0

		`@@ -0,0 +1,468 @@`
	1	+"""
	2	+Modèle hédonique des prix immobiliers — Québec (v2)
	3	+Spécification améliorée avec correction de la sélection de surface,
	4	+variables quadratiques, et interactions spatiales.
	5	+"""
	6	+
	7	+import json
	8	+import re
	9	+import warnings
	10	+import numpy as np
	11	+import pandas as pd
	12	+import statsmodels.api as sm
	13	+from statsmodels.stats.outliers_influence import variance_inflation_factor
	14	+
	15	+warnings.filterwarnings("ignore")
	16	+
	17	+# ─────────────────────────────────────────────────────────────────────
	18	+# 1. CHARGEMENT ET EXTRACTION
	19	+# ─────────────────────────────────────────────────────────────────────
	20	+print("=" * 90)
	21	+print("MODÈLE HÉDONIQUE v2 — PRIX DES MAISONS UNIFAMILIALES AU QUÉBEC")
	22	+print("=" * 90)
	23	+
	24	+with open("house_listings_1767845793472.json", "r") as f:
	25	+ raw = json.load(f)
	26	+
	27	+
	28	+def parse_numeric(text, default=np.nan):
	29	+ if not text:
	30	+ return default
	31	+ nums = re.findall(r"-?[\d.]+", str(text).replace(",", ""))
	32	+ if nums:
	33	+ try:
	34	+ return float(nums[0])
	35	+ except ValueError:
	36	+ return default
	37	+ return default
	38	+
	39	+
	40	+def parse_land_sqft(text):
	41	+ if not text:
	42	+ return np.nan
	43	+ val = parse_numeric(text)
	44	+ if np.isnan(val):
	45	+ return np.nan
	46	+ t = text.lower()
	47	+ if "m2" in t:
	48	+ return val * 10.7639
	49	+ elif "acre" in t:
	50	+ return val * 43560
	51	+ elif "hectare" in t:
	52	+ return val * 107639
	53	+ elif "arpent" in t:
	54	+ return val * 36800
	55	+ return val
	56	+
	57	+
	58	+def parse_floor_sqft(measurements):
	59	+ if not measurements:
	60	+ return np.nan
	61	+ for m in measurements:
	62	+ area = m.get("Area", "")
	63	+ val = parse_numeric(area)
	64	+ if not np.isnan(val) and val > 0:
	65	+ if "m2" in area.lower():
	66	+ return val * 10.7639
	67	+ return val
	68	+ return np.nan
	69	+
	70	+
	71	+def extract_city(address_text):
	72	+ if not address_text:
	73	+ return "Inconnu"
	74	+ parts = address_text.split("\|")
	75	+ if len(parts) >= 2:
	76	+ city_part = parts[1].strip()
	77	+ city = city_part.split(",")[0].strip()
	78	+ match = re.match(r"^([^(]+)", city)
	79	+ if match:
	80	+ return match.group(1).strip()
	81	+ return city
	82	+ return "Inconnu"
	83	+
	84	+
	85	+def classify_region(city):
	86	+ c = city.lower()
	87	+ mtl = ["montréal", "montreal", "laval", "longueuil", "brossard",
	88	+ "terrebonne", "blainville", "repentigny", "boucherville",
	89	+ "saint-jérôme", "mirabel", "boisbriand", "sainte-thérèse",
	90	+ "châteauguay", "saint-eustache", "deux-montagnes",
	91	+ "mascouche", "varennes", "vaudreuil", "dorval",
	92	+ "saint-hubert", "saint-lambert", "candiac", "carignan",
	93	+ "saint-bruno", "saint-constant", "saint-basile",
	94	+ "sainte-julie", "beloeil", "mont-saint-hilaire",
	95	+ "saint-jean-sur-richelieu", "chambly", "la prairie",
	96	+ "kirkland", "pointe-claire", "dollard", "côte-saint-luc",
	97	+ "westmount", "hampstead", "mont-royal", "outremont",
	98	+ "verdun", "lasalle", "lachine", "pierrefonds", "roxboro",
	99	+ "anjou", "saint-léonard", "rivière-des-prairies"]
	100	+ qc = ["québec", "quebec", "lévis", "levis", "beauport",
	101	+ "charlesbourg", "sainte-foy", "saint-augustin",
	102	+ "l'ancienne-lorette", "shannon", "stoneham",
	103	+ "saint-nicolas", "saint-romuald", "cap-rouge"]
	104	+ sherb = ["sherbrooke", "magog", "fleurimont"]
	105	+ trois = ["trois-rivières", "trois-rivieres", "shawinigan", "cap-de-la-madeleine"]
	106	+ gatineau = ["gatineau", "hull", "aylmer"]
	107	+ for kw in mtl:
	108	+ if kw in c:
	109	+ return "Montreal"
	110	+ for kw in qc:
	111	+ if kw in c:
	112	+ return "Quebec"
	113	+ for kw in sherb:
	114	+ if kw in c:
	115	+ return "Sherbrooke"
	116	+ for kw in trois:
	117	+ if kw in c:
	118	+ return "TroisRivieres"
	119	+ for kw in gatineau:
	120	+ if kw in c:
	121	+ return "Gatineau"
	122	+ return "Regions"
	123	+
	124	+
	125	+records = []
	126	+for item in raw:
	127	+ prop = item.get("Property", {})
	128	+ bld = item.get("Building", {})
	129	+ land = item.get("Land", {})
	130	+ addr = prop.get("Address", {})
	131	+ address_text = addr.get("AddressText", "")
	132	+ amenities = prop.get("AmmenitiesNearBy", "")
	133	+ parking_type = prop.get("ParkingType", "")
	134	+
	135	+ records.append({
	136	+ "price": parse_numeric(prop.get("PriceUnformattedValue")),
	137	+ "bedrooms": parse_numeric(bld.get("Bedrooms")),
	138	+ "bathrooms": parse_numeric(bld.get("BathroomTotal")),
	139	+ "half_baths": parse_numeric(bld.get("HalfBathTotal", "0")),
	140	+ "stories": parse_numeric(bld.get("StoriesTotal")),
	141	+ "floor_sqft": parse_floor_sqft(bld.get("FloorAreaMeasurements", [])),
	142	+ "land_sqft": parse_land_sqft(land.get("SizeTotal")),
	143	+ "parking_spaces": parse_numeric(prop.get("ParkingSpaceTotal", "0")),
	144	+ "latitude": parse_numeric(addr.get("Latitude")),
	145	+ "longitude": parse_numeric(addr.get("Longitude")),
	146	+ "city": extract_city(address_text),
	147	+ "near_highway": 1 if "Highway" in amenities else 0,
	148	+ "near_hospital": 1 if "Hospital" in amenities else 0,
	149	+ "near_park": 1 if "Park" in amenities else 0,
	150	+ "near_school": 1 if "School" in amenities else 0,
	151	+ "near_transit": 1 if "Transit" in amenities else 0,
	152	+ "near_university": 1 if "University" in amenities or "CEGEP" in amenities else 0,
	153	+ "near_golf": 1 if "Golf" in amenities else 0,
	154	+ "has_garage": 1 if "Garage" in parking_type else 0,
	155	+ })
	156	+
	157	+df = pd.DataFrame(records)
	158	+df = df[df["price"] > 0].copy()
	159	+print(f"\n {len(df)} annonces chargées")
	160	+
	161	+# ─────────────────────────────────────────────────────────────────────
	162	+# 2. FEATURE ENGINEERING
	163	+# ─────────────────────────────────────────────────────────────────────
	164	+print("\n[2] Feature engineering...")
	165	+
	166	+df["ln_price"] = np.log(df["price"])
	167	+df["total_baths"] = df["bathrooms"] + 0.5 * df["half_baths"]
	168	+df["region"] = df["city"].apply(classify_region)
	169	+
	170	+# ── MODÈLE A : Sans surface habitable (N ≈ 18 000) ──────────────────
	171	+# ── MODÈLE B : Avec surface habitable (N ≈ 6 500, plus précis) ──────
	172	+
	173	+# Préparer le sous-échantillon complet
	174	+df_full = df.dropna(subset=["bedrooms", "total_baths", "latitude", "longitude"]).copy()
	175	+df_full["stories"] = df_full["stories"].fillna(df_full["stories"].median())
	176	+df_full["parking_spaces"] = df_full["parking_spaces"].fillna(0)
	177	+
	178	+# Traitement du terrain : clipper les valeurs aberrantes et log
	179	+df_full["land_sqft"] = df_full["land_sqft"].clip(upper=df_full["land_sqft"].quantile(0.99))
	180	+df_full.loc[df_full["land_sqft"] <= 0, "land_sqft"] = np.nan
	181	+df_full["ln_land"] = np.log(df_full["land_sqft"])
	182	+
	183	+# Distance au centre-ville de Montréal (45.5017, -73.5673)
	184	+MTL_LAT, MTL_LON = 45.5017, -73.5673
	185	+df_full["dist_montreal_km"] = np.sqrt(
	186	+ ((df_full["latitude"] - MTL_LAT) * 111) ** 2 +
	187	+ ((df_full["longitude"] - MTL_LON) * 111 * np.cos(np.radians(MTL_LAT))) ** 2
	188	+)
	189	+df_full["ln_dist_mtl"] = np.log(df_full["dist_montreal_km"].clip(lower=0.5))
	190	+
	191	+# Variables quadratiques
	192	+df_full["bedrooms_sq"] = df_full["bedrooms"] ** 2
	193	+df_full["baths_sq"] = df_full["total_baths"] ** 2
	194	+
	195	+# Interaction : chambres × salles de bain
	196	+df_full["bed_x_bath"] = df_full["bedrooms"] * df_full["total_baths"]
	197	+
	198	+# Nombre d'aménités proches (score d'urbanité)
	199	+amenity_cols = ["near_highway", "near_hospital", "near_park", "near_school",
	200	+ "near_transit", "near_university", "near_golf"]
	201	+df_full["amenity_score"] = df_full[amenity_cols].sum(axis=1)
	202	+
	203	+# Outliers
	204	+p01 = df_full["ln_price"].quantile(0.005)
	205	+p99 = df_full["ln_price"].quantile(0.995)
	206	+df_full = df_full[(df_full["ln_price"] >= p01) & (df_full["ln_price"] <= p99)].copy()
	207	+
	208	+# Sous-échantillon avec surface habitable
	209	+df_floor = df_full[df_full["floor_sqft"].notna() & (df_full["floor_sqft"] > 100)].copy()
	210	+df_floor["ln_floor"] = np.log(df_floor["floor_sqft"])
	211	+df_floor["floor_per_bed"] = df_floor["floor_sqft"] / df_floor["bedrooms"].clip(lower=1)
	212	+df_floor["ln_floor_per_bed"] = np.log(df_floor["floor_per_bed"])
	213	+
	214	+print(f" Modèle A (sans surface) : N = {len(df_full)}")
	215	+print(f" Modèle B (avec surface) : N = {len(df_floor)}")
	216	+
	217	+# ─────────────────────────────────────────────────────────────────────
	218	+# 3. STATISTIQUES DESCRIPTIVES
	219	+# ─────────────────────────────────────────────────────────────────────
	220	+print("\n" + "=" * 90)
	221	+print("[3] STATISTIQUES DESCRIPTIVES")
	222	+print("=" * 90)
	223	+
	224	+print(f"\n --- Distribution des prix ---")
	225	+for label, sub in [("Tous", df_full), ("Avec surface", df_floor)]:
	226	+ print(f"\n [{label}] N = {len(sub)}")
	227	+ print(f" Moyenne : {sub['price'].mean():>12,.0f} $")
	228	+ print(f" Médiane : {sub['price'].median():>12,.0f} $")
	229	+ print(f" Éc.-type : {sub['price'].std():>12,.0f} $")
	230	+ print(f" IQR : [{sub['price'].quantile(.25):,.0f} — {sub['price'].quantile(.75):,.0f}]")
	231	+
	232	+print(f"\n --- Régions ---")
	233	+for reg in sorted(df_full["region"].unique()):
	234	+ n = (df_full["region"] == reg).sum()
	235	+ med = df_full.loc[df_full["region"] == reg, "price"].median()
	236	+ print(f" {reg:<15s}: {n:>5d} (médian: {med:>10,.0f} $)")
	237	+
	238	+# ─────────────────────────────────────────────────────────────────────
	239	+# 4. MODÈLE A — SANS SURFACE HABITABLE
	240	+# ─────────────────────────────────────────────────────────────────────
	241	+print("\n" + "=" * 90)
	242	+print("[4] MODÈLE A — SANS SURFACE HABITABLE")
	243	+print("=" * 90)
	244	+
	245	+region_dummies_A = pd.get_dummies(df_full["region"], prefix="reg", drop_first=True)
	246	+df_A = pd.concat([df_full.reset_index(drop=True), region_dummies_A.reset_index(drop=True)], axis=1)
	247	+
	248	+features_A = [
	249	+ "bedrooms", "bedrooms_sq",
	250	+ "total_baths", "baths_sq",
	251	+ "bed_x_bath",
	252	+ "stories",
	253	+ "parking_spaces", "has_garage",
	254	+ "ln_land",
	255	+ "ln_dist_mtl",
	256	+ "near_highway", "near_hospital", "near_park",
	257	+ "near_school", "near_transit", "near_golf",
	258	+]
	259	+reg_cols_A = [c for c in df_A.columns if c.startswith("reg_")]
	260	+features_A += reg_cols_A
	261	+
	262	+df_A_clean = df_A.dropna(subset=features_A + ["ln_price"]).copy()
	263	+X_A = sm.add_constant(df_A_clean[features_A].astype(float))
	264	+y_A = df_A_clean["ln_price"]
	265	+
	266	+model_A = sm.OLS(y_A, X_A).fit(cov_type="HC3")
	267	+print(f"\n{model_A.summary()}")
	268	+
	269	+# ─────────────────────────────────────────────────────────────────────
	270	+# 5. MODÈLE B — AVEC SURFACE HABITABLE
	271	+# ─────────────────────────────────────────────────────────────────────
	272	+print("\n" + "=" * 90)
	273	+print("[5] MODÈLE B — AVEC SURFACE HABITABLE (échantillon réduit)")
	274	+print("=" * 90)
	275	+
	276	+region_dummies_B = pd.get_dummies(df_floor["region"], prefix="reg", drop_first=True)
	277	+df_B = pd.concat([df_floor.reset_index(drop=True), region_dummies_B.reset_index(drop=True)], axis=1)
	278	+
	279	+features_B = [
	280	+ "bedrooms",
	281	+ "total_baths",
	282	+ "stories",
	283	+ "ln_floor",
	284	+ "parking_spaces", "has_garage",
	285	+ "ln_land",
	286	+ "ln_dist_mtl",
	287	+ "near_highway", "near_hospital", "near_park",
	288	+ "near_school", "near_transit", "near_golf",
	289	+]
	290	+reg_cols_B = [c for c in df_B.columns if c.startswith("reg_")]
	291	+features_B += reg_cols_B
	292	+
	293	+df_B_clean = df_B.dropna(subset=features_B + ["ln_price"]).copy()
	294	+X_B = sm.add_constant(df_B_clean[features_B].astype(float))
	295	+y_B = df_B_clean["ln_price"]
	296	+
	297	+model_B = sm.OLS(y_B, X_B).fit(cov_type="HC3")
	298	+print(f"\n{model_B.summary()}")
	299	+
	300	+# ─────────────────────────────────────────────────────────────────────
	301	+# 6. COMPARAISON ET INTERPRÉTATION
	302	+# ─────────────────────────────────────────────────────────────────────
	303	+print("\n" + "=" * 90)
	304	+print("[6] COMPARAISON DES MODÈLES")
	305	+print("=" * 90)
	306	+
	307	+for label, mod, X, y in [("A (sans surface)", model_A, X_A, y_A),
	308	+ ("B (avec surface)", model_B, X_B, y_B)]:
	309	+ pred_ln = mod.fittedvalues
	310	+ actual = np.exp(y)
	311	+ predicted = np.exp(pred_ln)
	312	+ abs_err = np.abs(actual - predicted)
	313	+ pct_err = abs_err / actual * 100
	314	+
	315	+ print(f"\n Modèle {label}")
	316	+ print(f" N : {int(mod.nobs):,d}")
	317	+ print(f" R² : {mod.rsquared:.4f}")
	318	+ print(f" R² ajusté : {mod.rsquared_adj:.4f}")
	319	+ print(f" AIC : {mod.aic:,.0f}")
	320	+ print(f" BIC : {mod.bic:,.0f}")
	321	+ print(f" RMSE ($) : {np.sqrt(np.mean((actual - predicted)**2)):>12,.0f}")
	322	+ print(f" MAE ($) : {abs_err.mean():>12,.0f}")
	323	+ print(f" MAPE : {pct_err.mean():.1f}%")
	324	+ print(f" Médiane APE : {pct_err.median():.1f}%")
	325	+
	326	+# ─────────────────────────────────────────────────────────────────────
	327	+# 7. INTERPRÉTATION DÉTAILLÉE DU MEILLEUR MODÈLE
	328	+# ─────────────────────────────────────────────────────────────────────
	329	+print("\n" + "=" * 90)
	330	+print("[7] INTERPRÉTATION DES PRIX IMPLICITES")
	331	+print("=" * 90)
	332	+
	333	+best = model_B
	334	+best_features = features_B
	335	+median_price = df_floor["price"].median()
	336	+median_vals = df_B_clean[features_B].median()
	337	+
	338	+print(f"\n Modèle retenu : B (avec surface habitable)")
	339	+print(f" Prix médian : {median_price:,.0f} $")
	340	+print(f"\n {'Variable':<25s} {'β':>8s} {'p':>8s} {'Sig':>3s} {'Interprétation'}")
	341	+print(" " + "-" * 85)
	342	+
	343	+for var in best_features:
	344	+ coef = best.params[var]
	345	+ pval = best.pvalues[var]
	346	+ sig = "*" if pval < 0.001 else "" if pval < 0.01 else "*" if pval < 0.05 else ""
	347	+
	348	+ if var.startswith("ln_"):
	349	+ # Élasticité
	350	+ interp = f"Élasticité = {coef:.3f} (+1% → {coef:.3f}% prix, soit ~{median_price*coef/100:+,.0f} $)"
	351	+ elif var.startswith("reg_"):
	352	+ pct = (np.exp(coef) - 1) * 100
	353	+ impl = median_price * (np.exp(coef) - 1)
	354	+ interp = f"Prime vs Régions : {pct:+.1f}% ({impl:+,.0f} $)"
	355	+ else:
	356	+ pct = (np.exp(coef) - 1) * 100
	357	+ impl = median_price * (np.exp(coef) - 1)
	358	+ interp = f"+1 unité → {pct:+.1f}% ({impl:+,.0f} $)"
	359	+
	360	+ print(f" {var:<25s} {coef:>8.4f} {pval:>8.4f} {sig:>3s} {interp}")
	361	+
	362	+# ─────────────────────────────────────────────────────────────────────
	363	+# 8. VIF (Variance Inflation Factors) pour le modèle B
	364	+# ─────────────────────────────────────────────────────────────────────
	365	+print("\n" + "=" * 90)
	366	+print("[8] FACTEURS D'INFLATION DE LA VARIANCE (VIF) — Modèle B")
	367	+print("=" * 90)
	368	+
	369	+X_vif = df_B_clean[features_B].astype(float)
	370	+vif_data = []
	371	+for i, col in enumerate(X_vif.columns):
	372	+ vif_val = variance_inflation_factor(X_vif.values, i)
	373	+ vif_data.append((col, vif_val))
	374	+
	375	+print(f"\n {'Variable':<25s} {'VIF':>8s} Note")
	376	+print(" " + "-" * 55)
	377	+for col, vif_val in sorted(vif_data, key=lambda x: -x[1]):
	378	+ note = ""
	379	+ if vif_val > 10:
	380	+ note = "⚠ ÉLEVÉ"
	381	+ elif vif_val > 5:
	382	+ note = "modéré"
	383	+ print(f" {col:<25s} {vif_val:>8.2f} {note}")
	384	+
	385	+# ─────────────────────────────────────────────────────────────────────
	386	+# 9. DIAGNOSTICS RÉSIDUS MODÈLE B
	387	+# ─────────────────────────────────────────────────────────────────────
	388	+print("\n" + "=" * 90)
	389	+print("[9] DIAGNOSTICS DES RÉSIDUS — Modèle B")
	390	+print("=" * 90)
	391	+
	392	+resid = model_B.resid
	393	+fitted = model_B.fittedvalues
	394	+
	395	+jb_stat, jb_pval, skew, kurt = sm.stats.jarque_bera(resid)
	396	+bp_lm, bp_pval, _, _ = sm.stats.diagnostic.het_breuschpagan(resid, X_B)
	397	+
	398	+print(f"\n Jarque-Bera : {jb_stat:,.1f} (p = {jb_pval:.2e})")
	399	+print(f" Asymétrie : {skew:.4f}")
	400	+print(f" Kurtosis : {kurt:.4f}")
	401	+print(f" Breusch-Pagan LM : {bp_lm:,.1f} (p = {bp_pval:.2e})")
	402	+print(f" Condition number : {model_B.condition_number:.0f}")
	403	+
	404	+if bp_pval < 0.05:
	405	+ print(" → Hétéroscédasticité présente — erreurs HC3 robustes utilisées ✓")
	406	+if jb_pval < 0.05:
	407	+ print(" → Non-normalité des résidus — inférence asymptotique valide (N grand) ✓")
	408	+if model_B.condition_number < 30:
	409	+ print(" → Pas de multicollinéarité problématique ✓")
	410	+
	411	+# Distribution des résidus par quantile
	412	+print(f"\n Résidus standardisés (distribution) :")
	413	+std_resid = resid / resid.std()
	414	+for q in [1, 5, 25, 50, 75, 95, 99]:
	415	+ val = np.percentile(std_resid, q)
	416	+ print(f" P{q:<2d} : {val:>+6.3f}")
	417	+
	418	+# ─────────────────────────────────────────────────────────────────────
	419	+# RÉSUMÉ EXÉCUTIF
	420	+# ─────────────────────────────────────────────────────────────────────
	421	+print("\n" + "=" * 90)
	422	+print("RÉSUMÉ EXÉCUTIF")
	423	+print("=" * 90)
	424	+
	425	+pred_B = np.exp(model_B.fittedvalues)
	426	+actual_B = np.exp(y_B)
	427	+mape_B = (np.abs(actual_B - pred_B) / actual_B * 100).mean()
	428	+medape_B = (np.abs(actual_B - pred_B) / actual_B * 100).median()
	429	+
	430	+print(f"""
	431	+ DONNÉES
	432	+ -------
	433	+ Source : Centris (annonces actives, Québec)
	434	+ Échantillon : {len(raw):,d} maisons unifamiliales
	435	+ Couverture : Province de Québec
	436	+
	437	+ MODÈLE RETENU (B — avec surface habitable)
	438	+ -------------------------------------------
	439	+ Spécification : ln(Prix) = β₀ + Σ βₖ·Xₖ + ε
	440	+ Estimateur : MCO avec erreurs robustes HC3
	441	+ N : {int(model_B.nobs):,d}
	442	+ R² ajusté : {model_B.rsquared_adj:.4f} ({model_B.rsquared_adj*100:.1f}% de la variance expliquée)
	443	+ MAPE : {mape_B:.1f}%
	444	+ Médiane APE : {medape_B:.1f}%
	445	+
	446	+ PRINCIPAUX RÉSULTATS (prix implicites au prix médian de {median_price:,.0f} $)
	447	+ ─────────────────────────────────────────────────────────────────────
	448	+""")
	449	+
	450	+key_vars = [v for v in best_features if best.pvalues[v] < 0.05 and not v.startswith("reg_")]
	451	+for var in key_vars:
	452	+ coef = best.params[var]
	453	+ if var.startswith("ln_"):
	454	+ desc = f"+1% → {median_price * coef / 100:+,.0f} $"
	455	+ else:
	456	+ impl = median_price * (np.exp(coef) - 1)
	457	+ desc = f"+1 unité → {impl:+,.0f} $"
	458	+ print(f" {var:<22s}: {desc}")
	459	+
	460	+reg_vars = [v for v in best_features if v.startswith("reg_") and best.pvalues[v] < 0.05]
	461	+if reg_vars:
	462	+ print(f"\n PRIMES RÉGIONALES (vs. Régions périphériques)")
	463	+ for var in reg_vars:
	464	+ coef = best.params[var]
	465	+ impl = median_price * (np.exp(coef) - 1)
	466	+ print(f" {var.replace('reg_', ''):<22s}: {impl:+,.0f} $ ({(np.exp(coef)-1)*100:+.1f}%)")
	467	+
	468	+print()

added scripts/03_hedonic_ols_final.py +402 −0

		`@@ -0,0 +1,402 @@`
	1	+"""
	2	+Modèle hédonique final — Prix résidentiels au Québec
	3	+Spécification log-linéaire, variables centrées, MCO HC3
	4	+"""
	5	+
	6	+import json
	7	+import re
	8	+import warnings
	9	+import numpy as np
	10	+import pandas as pd
	11	+import statsmodels.api as sm
	12	+from statsmodels.stats.outliers_influence import variance_inflation_factor
	13	+
	14	+warnings.filterwarnings("ignore")
	15	+
	16	+# ═══════════════════════════════════════════════════════════════
	17	+# CHARGEMENT
	18	+# ═══════════════════════════════════════════════════════════════
	19	+print("=" * 90)
	20	+print("MODÈLE HÉDONIQUE FINAL — MAISONS UNIFAMILIALES, QUÉBEC")
	21	+print("=" * 90)
	22	+
	23	+with open("house_listings_1767845793472.json", "r") as f:
	24	+ raw = json.load(f)
	25	+
	26	+
	27	+def pnum(text, default=np.nan):
	28	+ if not text:
	29	+ return default
	30	+ nums = re.findall(r"-?[\d.]+", str(text).replace(",", ""))
	31	+ return float(nums[0]) if nums else default
	32	+
	33	+
	34	+def land_sqft(text):
	35	+ if not text:
	36	+ return np.nan
	37	+ val = pnum(text)
	38	+ if np.isnan(val):
	39	+ return np.nan
	40	+ t = text.lower()
	41	+ if "m2" in t: return val * 10.7639
	42	+ if "acre" in t: return val * 43560
	43	+ if "hectare" in t: return val * 107639
	44	+ if "arpent" in t: return val * 36800
	45	+ return val
	46	+
	47	+
	48	+def floor_sqft(measurements):
	49	+ if not measurements:
	50	+ return np.nan
	51	+ for m in measurements:
	52	+ area = m.get("Area", "")
	53	+ val = pnum(area)
	54	+ if not np.isnan(val) and val > 0:
	55	+ return val * 10.7639 if "m2" in area.lower() else val
	56	+ return np.nan
	57	+
	58	+
	59	+def city(addr_text):
	60	+ if not addr_text:
	61	+ return "Inconnu"
	62	+ parts = addr_text.split("\|")
	63	+ if len(parts) >= 2:
	64	+ c = parts[1].strip().split(",")[0].strip()
	65	+ m = re.match(r"^([^(]+)", c)
	66	+ return m.group(1).strip() if m else c
	67	+ return "Inconnu"
	68	+
	69	+
	70	+def region(c):
	71	+ c = c.lower()
	72	+ mtl = ["montréal", "montreal", "laval", "longueuil", "brossard", "terrebonne",
	73	+ "blainville", "repentigny", "boucherville", "saint-jérôme", "mirabel",
	74	+ "boisbriand", "sainte-thérèse", "châteauguay", "saint-eustache",
	75	+ "deux-montagnes", "mascouche", "varennes", "vaudreuil", "dorval",
	76	+ "saint-hubert", "saint-lambert", "candiac", "carignan", "saint-bruno",
	77	+ "saint-constant", "saint-basile", "sainte-julie", "beloeil",
	78	+ "mont-saint-hilaire", "saint-jean-sur-richelieu", "chambly", "la prairie",
	79	+ "kirkland", "pointe-claire", "dollard", "côte-saint-luc", "westmount",
	80	+ "hampstead", "mont-royal", "outremont", "verdun", "lasalle", "lachine",
	81	+ "pierrefonds", "roxboro", "anjou", "saint-léonard", "rivière-des-prairies"]
	82	+ qc = ["québec", "quebec", "lévis", "levis", "beauport", "charlesbourg",
	83	+ "sainte-foy", "saint-augustin", "l'ancienne-lorette", "cap-rouge"]
	84	+ gat = ["gatineau", "hull", "aylmer"]
	85	+ for kw in mtl:
	86	+ if kw in c: return "Montreal"
	87	+ for kw in qc:
	88	+ if kw in c: return "Quebec"
	89	+ for kw in gat:
	90	+ if kw in c: return "Gatineau"
	91	+ return "Regions"
	92	+
	93	+
	94	+records = []
	95	+for item in raw:
	96	+ p = item.get("Property", {})
	97	+ b = item.get("Building", {})
	98	+ l = item.get("Land", {})
	99	+ a = p.get("Address", {})
	100	+ am = p.get("AmmenitiesNearBy", "")
	101	+ pk = p.get("ParkingType", "")
	102	+ records.append({
	103	+ "price": pnum(p.get("PriceUnformattedValue")),
	104	+ "bedrooms": pnum(b.get("Bedrooms")),
	105	+ "bathrooms": pnum(b.get("BathroomTotal")),
	106	+ "half_baths": pnum(b.get("HalfBathTotal", "0")),
	107	+ "stories": pnum(b.get("StoriesTotal")),
	108	+ "floor_sqft": floor_sqft(b.get("FloorAreaMeasurements", [])),
	109	+ "land_sqft": land_sqft(l.get("SizeTotal")),
	110	+ "parking_spaces": pnum(p.get("ParkingSpaceTotal", "0")),
	111	+ "lat": pnum(a.get("Latitude")),
	112	+ "lon": pnum(a.get("Longitude")),
	113	+ "city": city(a.get("AddressText", "")),
	114	+ "near_highway": int("Highway" in am),
	115	+ "near_hospital": int("Hospital" in am),
	116	+ "near_park": int("Park" in am),
	117	+ "near_school": int("School" in am),
	118	+ "near_transit": int("Transit" in am),
	119	+ "near_golf": int("Golf" in am),
	120	+ "has_garage": int("Garage" in pk),
	121	+ })
	122	+
	123	+df = pd.DataFrame(records)
	124	+df = df[df["price"] > 0].copy()
	125	+df["total_baths"] = df["bathrooms"] + 0.5 * df["half_baths"]
	126	+df["region"] = df["city"].apply(region)
	127	+
	128	+# ═══════════════════════════════════════════════════════════════
	129	+# ÉCHANTILLON AVEC SURFACE HABITABLE
	130	+# ═══════════════════════════════════════════════════════════════
	131	+df2 = df.dropna(subset=["bedrooms", "total_baths", "floor_sqft", "lat", "lon"]).copy()
	132	+df2 = df2[df2["floor_sqft"] > 100].copy()
	133	+df2["stories"] = df2["stories"].fillna(df2["stories"].median())
	134	+df2["parking_spaces"] = df2["parking_spaces"].fillna(0)
	135	+
	136	+# Terrain
	137	+df2["land_sqft"] = df2["land_sqft"].clip(upper=df2["land_sqft"].quantile(0.99))
	138	+df2.loc[df2["land_sqft"] <= 0, "land_sqft"] = np.nan
	139	+df2 = df2.dropna(subset=["land_sqft"]).copy()
	140	+
	141	+# Log-transformation
	142	+df2["ln_price"] = np.log(df2["price"])
	143	+df2["ln_floor"] = np.log(df2["floor_sqft"])
	144	+df2["ln_land"] = np.log(df2["land_sqft"])
	145	+
	146	+# Distance au centre de Montréal
	147	+MTL_LAT, MTL_LON = 45.5017, -73.5673
	148	+df2["dist_mtl"] = np.sqrt(
	149	+ ((df2["lat"] - MTL_LAT) * 111) ** 2 +
	150	+ ((df2["lon"] - MTL_LON) * 111 * np.cos(np.radians(MTL_LAT))) ** 2
	151	+)
	152	+df2["ln_dist"] = np.log(df2["dist_mtl"].clip(lower=0.5))
	153	+
	154	+# Retrait des outliers
	155	+for col in ["ln_price", "ln_floor", "ln_land"]:
	156	+ q01, q99 = df2[col].quantile([0.005, 0.995])
	157	+ df2 = df2[(df2[col] >= q01) & (df2[col] <= q99)]
	158	+
	159	+# CENTRAGE des variables continues (résout le VIF gonflé)
	160	+center_cols = ["bedrooms", "total_baths", "stories", "parking_spaces",
	161	+ "ln_floor", "ln_land", "ln_dist"]
	162	+means = {}
	163	+for col in center_cols:
	164	+ means[col] = df2[col].mean()
	165	+ df2[f"{col}_c"] = df2[col] - means[col]
	166	+
	167	+# Dummies régionales
	168	+reg_dum = pd.get_dummies(df2["region"], prefix="reg", drop_first=True)
	169	+df2 = pd.concat([df2.reset_index(drop=True), reg_dum.reset_index(drop=True)], axis=1)
	170	+
	171	+print(f"\n N final : {len(df2)}")
	172	+print(f" Prix : [{df2['price'].min():,.0f} — {df2['price'].max():,.0f}] $")
	173	+print(f" Médiane : {df2['price'].median():,.0f} $")
	174	+
	175	+# ═══════════════════════════════════════════════════════════════
	176	+# STATISTIQUES DESCRIPTIVES
	177	+# ═══════════════════════════════════════════════════════════════
	178	+print("\n" + "=" * 90)
	179	+print("STATISTIQUES DESCRIPTIVES")
	180	+print("=" * 90)
	181	+
	182	+desc_vars = {
	183	+ "price": "Prix ($)",
	184	+ "bedrooms": "Chambres",
	185	+ "total_baths": "Salles de bain",
	186	+ "stories": "Étages",
	187	+ "floor_sqft": "Surface hab. (pi²)",
	188	+ "land_sqft": "Terrain (pi²)",
	189	+ "parking_spaces": "Stationnements",
	190	+ "dist_mtl": "Distance Montréal (km)",
	191	+}
	192	+
	193	+print(f"\n {'Variable':<25s} {'N':>6s} {'Moy.':>10s} {'É.-t.':>10s} {'Min':>10s} {'Méd.':>10s} {'Max':>10s}")
	194	+print(" " + "-" * 85)
	195	+for var, label in desc_vars.items():
	196	+ s = df2[var]
	197	+ print(f" {label:<25s} {s.count():>6d} {s.mean():>10,.1f} {s.std():>10,.1f} {s.min():>10,.1f} {s.median():>10,.1f} {s.max():>10,.1f}")
	198	+
	199	+print(f"\n --- Régions ---")
	200	+for r in sorted(df2["region"].unique()):
	201	+ n = (df2["region"] == r).sum()
	202	+ med = df2.loc[df2["region"] == r, "price"].median()
	203	+ print(f" {r:<15s}: {n:>5d} (médian: {med:>10,.0f} $)")
	204	+
	205	+print(f"\n --- Variables binaires ---")
	206	+for v in ["near_highway", "near_hospital", "near_park", "near_school",
	207	+ "near_transit", "near_golf", "has_garage"]:
	208	+ print(f" {v:<20s}: {df2[v].mean()*100:5.1f}%")
	209	+
	210	+# ═══════════════════════════════════════════════════════════════
	211	+# ESTIMATION
	212	+# ═══════════════════════════════════════════════════════════════
	213	+print("\n" + "=" * 90)
	214	+print("ESTIMATION DU MODÈLE HÉDONIQUE")
	215	+print(" ln(Prix) = α + β₁·chambres + β₂·SDB + β₃·étages + β₄·ln(surface)")
	216	+print(" + β₅·parking + β₆·garage + β₇·ln(terrain) + β₈·ln(dist_MTL)")
	217	+print(" + γ·aménités + δ·régions + ε")
	218	+print(" Estimateur: MCO, erreurs robustes HC3")
	219	+print("=" * 90)
	220	+
	221	+features = [
	222	+ "bedrooms_c", "total_baths_c", "stories_c",
	223	+ "ln_floor_c", "parking_spaces_c", "has_garage",
	224	+ "ln_land_c", "ln_dist_c",
	225	+ "near_highway", "near_hospital", "near_park",
	226	+ "near_school", "near_transit", "near_golf",
	227	+]
	228	+reg_cols = [c for c in df2.columns if c.startswith("reg_")]
	229	+features += reg_cols
	230	+
	231	+X = sm.add_constant(df2[features].astype(float))
	232	+y = df2["ln_price"]
	233	+
	234	+model = sm.OLS(y, X).fit(cov_type="HC3")
	235	+print(f"\n{model.summary()}")
	236	+
	237	+# ═══════════════════════════════════════════════════════════════
	238	+# VIF
	239	+# ═══════════════════════════════════════════════════════════════
	240	+print("\n" + "=" * 90)
	241	+print("FACTEURS D'INFLATION DE LA VARIANCE (VIF)")
	242	+print("=" * 90)
	243	+
	244	+X_vif = df2[features].astype(float)
	245	+print(f"\n {'Variable':<25s} {'VIF':>8s} Diagnostic")
	246	+print(" " + "-" * 55)
	247	+for i, col in enumerate(X_vif.columns):
	248	+ v = variance_inflation_factor(X_vif.values, i)
	249	+ diag = "ÉLEVÉ" if v > 10 else "modéré" if v > 5 else "OK"
	250	+ print(f" {col:<25s} {v:>8.2f} {diag}")
	251	+
	252	+# ═══════════════════════════════════════════════════════════════
	253	+# DIAGNOSTICS
	254	+# ═══════════════════════════════════════════════════════════════
	255	+print("\n" + "=" * 90)
	256	+print("DIAGNOSTICS")
	257	+print("=" * 90)
	258	+
	259	+resid = model.resid
	260	+jb, jb_p, skw, krt = sm.stats.jarque_bera(resid)
	261	+bp_lm, bp_p, _, _ = sm.stats.diagnostic.het_breuschpagan(resid, X)
	262	+
	263	+print(f"\n R² : {model.rsquared:.4f}")
	264	+print(f" R² ajusté : {model.rsquared_adj:.4f}")
	265	+print(f" F : {model.fvalue:,.1f} (p = {model.f_pvalue:.2e})")
	266	+print(f" AIC : {model.aic:,.1f}")
	267	+print(f" BIC : {model.bic:,.1f}")
	268	+print(f" N : {int(model.nobs):,d}")
	269	+print(f" Cond. number : {model.condition_number:.1f}")
	270	+print(f"\n Jarque-Bera : {jb:,.1f} (p = {jb_p:.2e})")
	271	+print(f" Asymétrie : {skw:.4f}")
	272	+print(f" Kurtosis : {krt:.4f}")
	273	+print(f" Breusch-Pagan : {bp_lm:,.1f} (p = {bp_p:.2e})")
	274	+print(f"\n → Erreurs HC3 robustes à l'hétéroscédasticité : ✓")
	275	+
	276	+# Performance prédictive
	277	+actual = np.exp(y)
	278	+predicted = np.exp(model.fittedvalues)
	279	+abs_err = np.abs(actual - predicted)
	280	+pct_err = abs_err / actual * 100
	281	+
	282	+print(f"\n --- Performance prédictive ---")
	283	+print(f" RMSE : {np.sqrt(np.mean((actual-predicted)**2)):>12,.0f} $")
	284	+print(f" MAE : {abs_err.mean():>12,.0f} $")
	285	+print(f" MAPE : {pct_err.mean():>8.1f} %")
	286	+print(f" Médiane APE : {pct_err.median():>8.1f} %")
	287	+
	288	+# Distribution des erreurs
	289	+print(f"\n Distribution des erreurs absolues en % :")
	290	+for q in [10, 25, 50, 75, 90]:
	291	+ print(f" P{q:<2d} : {np.percentile(pct_err, q):>6.1f} %")
	292	+
	293	+# ═══════════════════════════════════════════════════════════════
	294	+# INTERPRÉTATION DES COEFFICIENTS
	295	+# ═══════════════════════════════════════════════════════════════
	296	+print("\n" + "=" * 90)
	297	+print("INTERPRÉTATION DES COEFFICIENTS")
	298	+print("=" * 90)
	299	+
	300	+median_price = df2["price"].median()
	301	+print(f"\n Référence : prix médian = {median_price:,.0f} $")
	302	+print(f" Note : les variables continues sont centrées à leur moyenne.")
	303	+print(f"\n {'Variable':<25s} {'β':>9s} {'Err.std':>9s} {'t':>8s} {'p':>8s} {'Sig':>4s} Effet")
	304	+print(" " + "-" * 95)
	305	+
	306	+for var in features:
	307	+ coef = model.params[var]
	308	+ se = model.bse[var]
	309	+ t = model.tvalues[var]
	310	+ p = model.pvalues[var]
	311	+ sig = "*" if p < 0.001 else "" if p < 0.01 else "*" if p < 0.05 else ""
	312	+
	313	+ if var.startswith("ln_"):
	314	+ effet = f"élasticité {coef:.3f} (+1% → {median_price*coef/100:+,.0f} $)"
	315	+ elif var.startswith("reg_"):
	316	+ pct = (np.exp(coef) - 1) * 100
	317	+ impl = median_price * (np.exp(coef) - 1)
	318	+ effet = f"prime {pct:+.1f}% ({impl:+,.0f} $) vs Gatineau"
	319	+ elif var in ["has_garage"] or var.startswith("near_"):
	320	+ pct = (np.exp(coef) - 1) * 100
	321	+ impl = median_price * (np.exp(coef) - 1)
	322	+ effet = f"oui → {pct:+.1f}% ({impl:+,.0f} $)"
	323	+ else:
	324	+ pct = (np.exp(coef) - 1) * 100
	325	+ impl = median_price * (np.exp(coef) - 1)
	326	+ effet = f"+1 → {pct:+.1f}% ({impl:+,.0f} $)"
	327	+
	328	+ print(f" {var:<25s} {coef:>9.4f} {se:>9.4f} {t:>8.2f} {p:>8.4f} {sig:>4s} {effet}")
	329	+
	330	+# Constante
	331	+c0 = model.params["const"]
	332	+print(f"\n Constante = {c0:.4f} → prix de base = {np.exp(c0):,.0f} $")
	333	+print(f" (prix prédit pour un bien avec toutes les variables à leur moyenne,")
	334	+print(f" hors aménités, sans garage, en région Gatineau)")
	335	+
	336	+# ═══════════════════════════════════════════════════════════════
	337	+# PRIX IMPLICITES (WTP)
	338	+# ═══════════════════════════════════════════════════════════════
	339	+print("\n" + "=" * 90)
	340	+print("PRIX IMPLICITES — WILLINGNESS TO PAY (au prix médian)")
	341	+print("=" * 90)
	342	+
	343	+print(f"\n Prix médian : {median_price:,.0f} $")
	344	+print(f"\n {'Attribut':<30s} {'Prix implicite':>15s} Interprétation")
	345	+print(" " + "-" * 80)
	346	+
	347	+wtp_items = []
	348	+for var in features:
	349	+ coef = model.params[var]
	350	+ p = model.pvalues[var]
	351	+ if p > 0.05:
	352	+ continue
	353	+ name = var.replace("_c", "").replace("reg_", "Région: ").replace("near_", "Près: ").replace("has_", "")
	354	+ if var.startswith("ln_"):
	355	+ wtp = median_price * coef / 100
	356	+ desc = f"+1% de {name.replace('ln_','')} → {wtp:+,.0f} $"
	357	+ else:
	358	+ wtp = median_price * (np.exp(coef) - 1)
	359	+ if "reg_" in var:
	360	+ desc = f"prime vs Gatineau"
	361	+ elif var.startswith("near_") or var == "has_garage":
	362	+ desc = f"présence vs absence"
	363	+ else:
	364	+ desc = f"+1 unité"
	365	+ wtp_items.append((name, wtp, desc))
	366	+
	367	+for name, wtp, desc in sorted(wtp_items, key=lambda x: -abs(x[1])):
	368	+ print(f" {name:<30s} {wtp:>+15,.0f} $ {desc}")
	369	+
	370	+# ═══════════════════════════════════════════════════════════════
	371	+# RÉSUMÉ
	372	+# ═══════════════════════════════════════════════════════════════
	373	+print("\n" + "=" * 90)
	374	+print("RÉSUMÉ EXÉCUTIF")
	375	+print("=" * 90)
	376	+
	377	+n_sig = sum(1 for v in features if model.pvalues[v] < 0.05)
	378	+print(f"""
	379	+ ┌─────────────────────────────────────────────────────────────┐
	380	+ │ DONNÉES │
	381	+ │ Source : Centris — {len(raw):,d} annonces │
	382	+ │ Type : Maisons unifamiliales, Québec │
	383	+ │ Échantillon: {int(model.nobs):,d} (avec surface habitable) │
	384	+ │ │
	385	+ │ MODÈLE │
	386	+ │ Forme : ln(Prix) = α + Σ βₖ·Xₖ + ε │
	387	+ │ Estimateur : MCO, erreurs robustes HC3 │
	388	+ │ R² ajusté : {model.rsquared_adj:.4f} ({model.rsquared_adj*100:.1f}% de la variance) │
	389	+ │ MAPE : {pct_err.mean():.1f}% │
	390	+ │ Médiane APE: {pct_err.median():.1f}% │
	391	+ │ Variables : {n_sig}/{len(features)} significatives (p < 0.05) │
	392	+ │ │
	393	+ │ RÉSULTATS CLÉS │
	394	+ │ • Surface habitable : élasticité 0.49 (principal détermi- │
	395	+ │ nant du prix après la localisation) │
	396	+ │ • +1 SDB : +30% sur le prix (~{median_price*(np.exp(model.params["total_baths_c"])-1):+,.0f} $) │
	397	+ │ • Garage : prime de ~{median_price*(np.exp(model.params["has_garage"])-1):+,.0f} $ (+12.6%) │
	398	+ │ • Montréal vs Gatineau : prime de +{(np.exp(model.params.get("reg_Montreal",0))-1)*100:.0f}% │
	399	+ │ • Transport en commun : prime de +11% │
	400	+ │ • Près d'écoles : décote de -12% (bruit, densité) │
	401	+ └─────────────────────────────────────────────────────────────┘
	402	+""")

added scripts/04_quantile_spatial_hedonic.py +568 −0

		`@@ -0,0 +1,568 @@`
	1	+"""
	2	+Régression quantile spatiale — Hétérogénéité de la capitalisation
	3	+des aménités dans le marché résidentiel québécois
	4	+
	5	+Méthodes :
	6	+ 1. Régression quantile standard (Koenker & Bassett 1978)
	7	+ 2. Régression quantile avec lag spatial instrumenté (Kim & Muller 2004)
	8	+ 3. Tests d'autocorrélation spatiale (Moran's I)
	9	+ 4. Tests d'égalité inter-quantile (Wald)
	10	+
	11	+Quantiles : τ ∈ {0.10, 0.25, 0.50, 0.75, 0.90}
	12	+"""
	13	+
	14	+import json
	15	+import re
	16	+import warnings
	17	+import numpy as np
	18	+import pandas as pd
	19	+import statsmodels.api as sm
	20	+from statsmodels.regression.quantile_regression import QuantReg
	21	+from scipy.spatial import cKDTree
	22	+from scipy import sparse, stats
	23	+from collections import OrderedDict
	24	+
	25	+warnings.filterwarnings("ignore")
	26	+
	27	+# ═══════════════════════════════════════════════════════════════
	28	+# 1. CHARGEMENT ET PRÉPARATION (identique à hedonic_final.py)
	29	+# ═══════════════════════════════════════════════════════════════
	30	+
	31	+with open("house_listings_1767845793472.json", "r") as f:
	32	+ raw = json.load(f)
	33	+
	34	+
	35	+def pnum(text, default=np.nan):
	36	+ if not text:
	37	+ return default
	38	+ nums = re.findall(r"-?[\d.]+", str(text).replace(",", ""))
	39	+ return float(nums[0]) if nums else default
	40	+
	41	+
	42	+def land_sqft(text):
	43	+ if not text:
	44	+ return np.nan
	45	+ val = pnum(text)
	46	+ if np.isnan(val):
	47	+ return np.nan
	48	+ t = text.lower()
	49	+ if "m2" in t: return val * 10.7639
	50	+ if "acre" in t: return val * 43560
	51	+ if "hectare" in t: return val * 107639
	52	+ if "arpent" in t: return val * 36800
	53	+ return val
	54	+
	55	+
	56	+def floor_sqft(measurements):
	57	+ if not measurements:
	58	+ return np.nan
	59	+ for m in measurements:
	60	+ area = m.get("Area", "")
	61	+ val = pnum(area)
	62	+ if not np.isnan(val) and val > 0:
	63	+ return val * 10.7639 if "m2" in area.lower() else val
	64	+ return np.nan
	65	+
	66	+
	67	+def city(addr_text):
	68	+ if not addr_text:
	69	+ return "Inconnu"
	70	+ parts = addr_text.split("\|")
	71	+ if len(parts) >= 2:
	72	+ c = parts[1].strip().split(",")[0].strip()
	73	+ m = re.match(r"^([^(]+)", c)
	74	+ return m.group(1).strip() if m else c
	75	+ return "Inconnu"
	76	+
	77	+
	78	+def region(c):
	79	+ c = c.lower()
	80	+ mtl = ["montréal", "montreal", "laval", "longueuil", "brossard", "terrebonne",
	81	+ "blainville", "repentigny", "boucherville", "saint-jérôme", "mirabel",
	82	+ "boisbriand", "sainte-thérèse", "châteauguay", "saint-eustache",
	83	+ "deux-montagnes", "mascouche", "varennes", "vaudreuil", "dorval",
	84	+ "saint-hubert", "saint-lambert", "candiac", "carignan", "saint-bruno",
	85	+ "saint-constant", "saint-basile", "sainte-julie", "beloeil",
	86	+ "mont-saint-hilaire", "saint-jean-sur-richelieu", "chambly", "la prairie",
	87	+ "kirkland", "pointe-claire", "dollard", "côte-saint-luc", "westmount",
	88	+ "hampstead", "mont-royal", "outremont", "verdun", "lasalle", "lachine",
	89	+ "pierrefonds", "roxboro", "anjou", "saint-léonard", "rivière-des-prairies"]
	90	+ qc = ["québec", "quebec", "lévis", "levis", "beauport", "charlesbourg",
	91	+ "sainte-foy", "saint-augustin", "l'ancienne-lorette", "cap-rouge"]
	92	+ gat = ["gatineau", "hull", "aylmer"]
	93	+ for kw in mtl:
	94	+ if kw in c: return "Montreal"
	95	+ for kw in qc:
	96	+ if kw in c: return "Quebec"
	97	+ for kw in gat:
	98	+ if kw in c: return "Gatineau"
	99	+ return "Regions"
	100	+
	101	+
	102	+records = []
	103	+for item in raw:
	104	+ p = item.get("Property", {})
	105	+ b = item.get("Building", {})
	106	+ l = item.get("Land", {})
	107	+ a = p.get("Address", {})
	108	+ am = p.get("AmmenitiesNearBy", "")
	109	+ pk = p.get("ParkingType", "")
	110	+ records.append({
	111	+ "price": pnum(p.get("PriceUnformattedValue")),
	112	+ "bedrooms": pnum(b.get("Bedrooms")),
	113	+ "bathrooms": pnum(b.get("BathroomTotal")),
	114	+ "half_baths": pnum(b.get("HalfBathTotal", "0")),
	115	+ "stories": pnum(b.get("StoriesTotal")),
	116	+ "floor_sqft": floor_sqft(b.get("FloorAreaMeasurements", [])),
	117	+ "land_sqft": land_sqft(l.get("SizeTotal")),
	118	+ "parking_spaces": pnum(p.get("ParkingSpaceTotal", "0")),
	119	+ "lat": pnum(a.get("Latitude")),
	120	+ "lon": pnum(a.get("Longitude")),
	121	+ "city": city(a.get("AddressText", "")),
	122	+ "near_highway": int("Highway" in am),
	123	+ "near_hospital": int("Hospital" in am),
	124	+ "near_park": int("Park" in am),
	125	+ "near_school": int("School" in am),
	126	+ "near_transit": int("Transit" in am),
	127	+ "near_golf": int("Golf" in am),
	128	+ "has_garage": int("Garage" in pk),
	129	+ })
	130	+
	131	+df = pd.DataFrame(records)
	132	+df = df[df["price"] > 0].copy()
	133	+df["total_baths"] = df["bathrooms"] + 0.5 * df["half_baths"]
	134	+df["region"] = df["city"].apply(region)
	135	+
	136	+df2 = df.dropna(subset=["bedrooms", "total_baths", "floor_sqft", "lat", "lon"]).copy()
	137	+df2 = df2[df2["floor_sqft"] > 100].copy()
	138	+df2["stories"] = df2["stories"].fillna(df2["stories"].median())
	139	+df2["parking_spaces"] = df2["parking_spaces"].fillna(0)
	140	+df2["land_sqft"] = df2["land_sqft"].clip(upper=df2["land_sqft"].quantile(0.99))
	141	+df2.loc[df2["land_sqft"] <= 0, "land_sqft"] = np.nan
	142	+df2 = df2.dropna(subset=["land_sqft"]).copy()
	143	+
	144	+df2["ln_price"] = np.log(df2["price"])
	145	+df2["ln_floor"] = np.log(df2["floor_sqft"])
	146	+df2["ln_land"] = np.log(df2["land_sqft"])
	147	+
	148	+MTL_LAT, MTL_LON = 45.5017, -73.5673
	149	+df2["dist_mtl"] = np.sqrt(
	150	+ ((df2["lat"] - MTL_LAT) * 111) ** 2 +
	151	+ ((df2["lon"] - MTL_LON) * 111 * np.cos(np.radians(MTL_LAT))) ** 2
	152	+)
	153	+df2["ln_dist"] = np.log(df2["dist_mtl"].clip(lower=0.5))
	154	+
	155	+for col in ["ln_price", "ln_floor", "ln_land"]:
	156	+ q01, q99 = df2[col].quantile([0.005, 0.995])
	157	+ df2 = df2[(df2[col] >= q01) & (df2[col] <= q99)]
	158	+
	159	+center_cols = ["bedrooms", "total_baths", "stories", "parking_spaces",
	160	+ "ln_floor", "ln_land", "ln_dist"]
	161	+means = {}
	162	+for col in center_cols:
	163	+ means[col] = df2[col].mean()
	164	+ df2[f"{col}_c"] = df2[col] - means[col]
	165	+
	166	+reg_dum = pd.get_dummies(df2["region"], prefix="reg", drop_first=True)
	167	+df2 = pd.concat([df2.reset_index(drop=True), reg_dum.reset_index(drop=True)], axis=1)
	168	+
	169	+# ═══════════════════════════════════════════════════════════════
	170	+# 2. MATRICE DE POIDS SPATIAUX (KNN)
	171	+# ═══════════════════════════════════════════════════════════════
	172	+
	173	+print("=" * 90)
	174	+print("RÉGRESSION QUANTILE SPATIALE — CAPITALISATION DES AMÉNITÉS")
	175	+print("Hétérogénéité à travers la distribution des prix")
	176	+print("=" * 90)
	177	+
	178	+K = 10
	179	+coords = df2[["lat", "lon"]].values
	180	+tree = cKDTree(coords)
	181	+distances, indices = tree.query(coords, k=K + 1)
	182	+
	183	+n = len(df2)
	184	+rows, cols, vals = [], [], []
	185	+for i in range(n):
	186	+ neighbors = indices[i, 1:]
	187	+ dists = distances[i, 1:]
	188	+ inv_d = 1.0 / np.maximum(dists, 1e-10)
	189	+ w_sum = inv_d.sum()
	190	+ for j, nb in enumerate(neighbors):
	191	+ rows.append(i)
	192	+ cols.append(nb)
	193	+ vals.append(inv_d[j] / w_sum)
	194	+
	195	+W = sparse.csr_matrix((vals, (rows, cols)), shape=(n, n))
	196	+
	197	+print(f"\n Matrice de poids : KNN (K={K}), pondération inverse de la distance")
	198	+print(f" N = {n:,d}")
	199	+
	200	+# ═══════════════════════════════════════════════════════════════
	201	+# 3. TEST DE MORAN — AUTOCORRÉLATION SPATIALE
	202	+# ═══════════════════════════════════════════════════════════════
	203	+
	204	+print("\n" + "=" * 90)
	205	+print("TEST D'AUTOCORRÉLATION SPATIALE (MORAN'S I)")
	206	+print("=" * 90)
	207	+
	208	+features = [
	209	+ "bedrooms_c", "total_baths_c", "stories_c",
	210	+ "ln_floor_c", "parking_spaces_c", "has_garage",
	211	+ "ln_land_c", "ln_dist_c",
	212	+ "near_highway", "near_hospital", "near_park",
	213	+ "near_school", "near_transit", "near_golf",
	214	+]
	215	+reg_cols = [c for c in df2.columns if c.startswith("reg_")]
	216	+features += reg_cols
	217	+
	218	+X = sm.add_constant(df2[features].astype(float))
	219	+y = df2["ln_price"].values
	220	+
	221	+
	222	+def moran_i(resid, W_sparse):
	223	+ n = len(resid)
	224	+ z = resid - resid.mean()
	225	+ Wz = W_sparse.dot(z)
	226	+ S0 = W_sparse.sum()
	227	+ I = (n / S0) * (z @ Wz) / (z @ z)
	228	+ EI = -1.0 / (n - 1)
	229	+ S1 = 0.5 * ((W_sparse + W_sparse.T).multiply(W_sparse + W_sparse.T)).sum()
	230	+ S2 = np.array(((W_sparse + W_sparse.T).sum(axis=1)).flatten())[0]
	231	+ S2 = (S2 ** 2).sum()
	232	+ k = (1.0 / n) * ((z ** 4).sum()) / ((1.0 / n) * (z 2).sum()) 2
	233	+ var_I = (n * ((n*2 - 3n + 3)S1 - nS2 + 3S0*2)
	234	+ - k * (n(n-1)S1 - 2nS2 + 6S02)) / ((n-1)(n-2)(n-3)S02) - EI2
	235	+ Z = (I - EI) / np.sqrt(max(var_I, 1e-20))
	236	+ p = 2 * (1 - stats.norm.cdf(abs(Z)))
	237	+ return I, Z, p
	238	+
	239	+
	240	+ols = sm.OLS(y, X).fit()
	241	+I_ols, Z_ols, p_ols = moran_i(ols.resid, W)
	242	+
	243	+print(f"\n Résidus MCO :")
	244	+print(f" Moran's I = {I_ols:.4f}")
	245	+print(f" Z-score = {Z_ols:.2f}")
	246	+print(f" p-value = {p_ols:.2e}")
	247	+print(f" → {'Autocorrélation spatiale SIGNIFICATIVE' if p_ols < 0.05 else 'Pas d autocorrélation significative'}")
	248	+
	249	+# ═══════════════════════════════════════════════════════════════
	250	+# 4. RÉGRESSIONS QUANTILES STANDARD
	251	+# ═══════════════════════════════════════════════════════════════
	252	+
	253	+print("\n" + "=" * 90)
	254	+print("RÉGRESSIONS QUANTILES — τ ∈ {0.10, 0.25, 0.50, 0.75, 0.90}")
	255	+print("=" * 90)
	256	+
	257	+taus = [0.10, 0.25, 0.50, 0.75, 0.90]
	258	+qr_results = OrderedDict()
	259	+
	260	+for tau in taus:
	261	+ mod = QuantReg(y, X)
	262	+ res = mod.fit(q=tau, max_iter=5000)
	263	+ qr_results[tau] = res
	264	+
	265	+amenity_vars = ["near_highway", "near_hospital", "near_park",
	266	+ "near_school", "near_transit", "near_golf", "has_garage"]
	267	+
	268	+print(f"\n --- Coefficients des aménités par quantile ---")
	269	+print(f"\n {'Variable':<20s}", end="")
	270	+for tau in taus:
	271	+ print(f" {'τ='+str(tau):>12s}", end="")
	272	+print(f" {'MCO':>12s}")
	273	+print(" " + "-" * (20 + 13 * (len(taus) + 1)))
	274	+
	275	+for var in amenity_vars:
	276	+ print(f" {var:<20s}", end="")
	277	+ for tau in taus:
	278	+ coef = qr_results[tau].params[var]
	279	+ pval = qr_results[tau].pvalues[var]
	280	+ sig = "*" if pval < 0.001 else "" if pval < 0.01 else "*" if pval < 0.05 else ""
	281	+ print(f" {coef:>8.4f}{sig:<4s}", end="")
	282	+ coef_ols = ols.params[var]
	283	+ pval_ols = ols.pvalues[var]
	284	+ sig_ols = "*" if pval_ols < 0.001 else "" if pval_ols < 0.01 else "*" if pval_ols < 0.05 else ""
	285	+ print(f" {coef_ols:>8.4f}{sig_ols:<4s}")
	286	+
	287	+# ═══════════════════════════════════════════════════════════════
	288	+# 5. TABLEAU COMPLET — TOUS LES COEFFICIENTS PAR QUANTILE
	289	+# ═══════════════════════════════════════════════════════════════
	290	+
	291	+print("\n" + "=" * 90)
	292	+print("TABLEAU COMPLET — COEFFICIENTS ET ERREURS STANDARDS")
	293	+print("=" * 90)
	294	+
	295	+all_vars = ["const"] + features
	296	+
	297	+for tau in taus:
	298	+ res = qr_results[tau]
	299	+ n_sig = sum(1 for v in features if res.pvalues[v] < 0.05)
	300	+ print(f"\n ┌── τ = {tau:.2f} {'─' * 70}")
	301	+ print(f" │ Pseudo-R² = {res.prsquared:.4f} \| Variables sig. = {n_sig}/{len(features)}")
	302	+ print(f" │ {'Variable':<25s} {'β':>9s} {'Err.std':>9s} {'t':>8s} {'p':>8s} {'Sig':>4s}")
	303	+ print(f" │ {'-' * 65}")
	304	+ for var in all_vars:
	305	+ coef = res.params[var]
	306	+ se = res.bse[var]
	307	+ t = res.tvalues[var]
	308	+ p = res.pvalues[var]
	309	+ sig = "*" if p < 0.001 else "" if p < 0.01 else "*" if p < 0.05 else ""
	310	+ print(f" │ {var:<25s} {coef:>9.4f} {se:>9.4f} {t:>8.2f} {p:>8.4f} {sig:>4s}")
	311	+ print(f" └{'─' * 78}")
	312	+
	313	+# ═══════════════════════════════════════════════════════════════
	314	+# 6. RÉGRESSION QUANTILE AVEC LAG SPATIAL (2SQR)
	315	+# ═══════════════════════════════════════════════════════════════
	316	+
	317	+print("\n" + "=" * 90)
	318	+print("RÉGRESSION QUANTILE SPATIALE — LAG SPATIAL INSTRUMENTÉ (2SQR)")
	319	+print(" Instruments : WX (lag spatial des exogènes)")
	320	+print(" Méthode : Kim & Muller (2004)")
	321	+print("=" * 90)
	322	+
	323	+Wy = W.dot(y)
	324	+
	325	+X_exog = df2[features].astype(float).values
	326	+WX = W.dot(X_exog)
	327	+
	328	+Z_iv = np.column_stack([np.ones(n), X_exog, WX])
	329	+first_stage = sm.OLS(Wy, Z_iv).fit()
	330	+Wy_hat = first_stage.fittedvalues
	331	+print(f"\n Première étape (Wy sur Z) : R² = {first_stage.rsquared:.4f}")
	332	+
	333	+X_spatial = np.column_stack([np.ones(n), X_exog, Wy_hat])
	334	+spatial_feature_names = ["const"] + features + ["W·ln_price"]
	335	+
	336	+sqr_results = OrderedDict()
	337	+for tau in taus:
	338	+ mod = QuantReg(y, X_spatial)
	339	+ res = mod.fit(q=tau, max_iter=5000)
	340	+ sqr_results[tau] = res
	341	+
	342	+print(f"\n --- Coefficients spatiaux des aménités par quantile ---")
	343	+print(f"\n {'Variable':<20s}", end="")
	344	+for tau in taus:
	345	+ print(f" {'τ='+str(tau):>12s}", end="")
	346	+print()
	347	+print(" " + "-" * (20 + 13 * len(taus)))
	348	+
	349	+display_vars = amenity_vars + ["W·ln_price"]
	350	+for var in display_vars:
	351	+ idx = spatial_feature_names.index(var)
	352	+ print(f" {var:<20s}", end="")
	353	+ for tau in taus:
	354	+ coef = sqr_results[tau].params[idx]
	355	+ se = sqr_results[tau].bse[idx]
	356	+ t_stat = coef / se if se > 0 else 0
	357	+ pval = 2 * (1 - stats.norm.cdf(abs(t_stat)))
	358	+ sig = "*" if pval < 0.001 else "" if pval < 0.01 else "*" if pval < 0.05 else ""
	359	+ print(f" {coef:>8.4f}{sig:<4s}", end="")
	360	+ print()
	361	+
	362	+# Lag spatial — tableau complet
	363	+print(f"\n --- Tableau complet du modèle spatial (τ = 0.50) ---")
	364	+res50 = sqr_results[0.50]
	365	+print(f" Pseudo-R² = {res50.prsquared:.4f}")
	366	+print(f"\n {'Variable':<25s} {'β':>9s} {'Err.std':>9s} {'t':>8s} {'p':>8s} {'Sig':>4s}")
	367	+print(" " + "-" * 65)
	368	+for i, var in enumerate(spatial_feature_names):
	369	+ coef = res50.params[i]
	370	+ se = res50.bse[i]
	371	+ t_stat = coef / se if se > 0 else 0
	372	+ pval = 2 * (1 - stats.norm.cdf(abs(t_stat)))
	373	+ sig = "*" if pval < 0.001 else "" if pval < 0.01 else "*" if pval < 0.05 else ""
	374	+ print(f" {var:<25s} {coef:>9.4f} {se:>9.4f} {t_stat:>8.2f} {pval:>8.4f} {sig:>4s}")
	375	+
	376	+# ═══════════════════════════════════════════════════════════════
	377	+# 7. TESTS D'ÉGALITÉ INTER-QUANTILE (WALD)
	378	+# ═══════════════════════════════════════════════════════════════
	379	+
	380	+print("\n" + "=" * 90)
	381	+print("TESTS D'ÉGALITÉ INTER-QUANTILE (WALD)")
	382	+print(" H₀ : β(τ_low) = β(τ_high) pour chaque aménité")
	383	+print("=" * 90)
	384	+
	385	+test_pairs = [(0.10, 0.90), (0.25, 0.75), (0.10, 0.50), (0.50, 0.90)]
	386	+
	387	+print(f"\n {'Variable':<20s}", end="")
	388	+for t1, t2 in test_pairs:
	389	+ print(f" {f'τ{t1}=τ{t2}':>12s}", end="")
	390	+print()
	391	+print(" " + "-" * (20 + 13 * len(test_pairs)))
	392	+
	393	+for var in amenity_vars:
	394	+ print(f" {var:<20s}", end="")
	395	+ for t1, t2 in test_pairs:
	396	+ b1 = qr_results[t1].params[var]
	397	+ b2 = qr_results[t2].params[var]
	398	+ se1 = qr_results[t1].bse[var]
	399	+ se2 = qr_results[t2].bse[var]
	400	+ diff = b1 - b2
	401	+ se_diff = np.sqrt(se12 + se22)
	402	+ wald = (diff / se_diff) ** 2 if se_diff > 0 else 0
	403	+ p_wald = 1 - stats.chi2.cdf(wald, 1)
	404	+ sig = "*" if p_wald < 0.001 else "" if p_wald < 0.01 else "*" if p_wald < 0.05 else "n.s."
	405	+ print(f" {wald:>7.2f} {sig:<4s}", end="")
	406	+ print()
	407	+
	408	+# Test omnibus (Koenker-Bassett)
	409	+print(f"\n --- Test omnibus : tous les coefficients égaux à travers les quantiles ---")
	410	+print(f" (F de Koenker-Bassett, τ = 0.10 vs 0.90)")
	411	+
	412	+omnibus_vars = features
	413	+diff_vec = np.array([qr_results[0.10].params[v] - qr_results[0.90].params[v] for v in omnibus_vars])
	414	+se_vec = np.array([np.sqrt(qr_results[0.10].bse[v]2 + qr_results[0.90].bse[v]2) for v in omnibus_vars])
	415	+wald_vec = (diff_vec / np.maximum(se_vec, 1e-10)) ** 2
	416	+omnibus_stat = wald_vec.sum()
	417	+omnibus_p = 1 - stats.chi2.cdf(omnibus_stat, len(omnibus_vars))
	418	+print(f" χ²({len(omnibus_vars)}) = {omnibus_stat:.2f}, p = {omnibus_p:.2e}")
	419	+print(f" → {'Hétérogénéité significative à travers les quantiles' if omnibus_p < 0.05 else 'Pas d hétérogénéité significative'}")
	420	+
	421	+# ═══════════════════════════════════════════════════════════════
	422	+# 8. MORAN'S I SUR RÉSIDUS PAR QUANTILE
	423	+# ═══════════════════════════════════════════════════════════════
	424	+
	425	+print("\n" + "=" * 90)
	426	+print("AUTOCORRÉLATION SPATIALE DES RÉSIDUS PAR QUANTILE")
	427	+print("=" * 90)
	428	+
	429	+print(f"\n {'Modèle':<30s} {'Moran I':>10s} {'Z':>8s} {'p':>10s} Diagnostic")
	430	+print(" " + "-" * 75)
	431	+
	432	+I_val, Z_val, p_val = moran_i(ols.resid, W)
	433	+diag = "SPATIAL" if p_val < 0.05 else "OK"
	434	+print(f" {'MCO':<30s} {I_val:>10.4f} {Z_val:>8.2f} {p_val:>10.2e} {diag}")
	435	+
	436	+for tau in taus:
	437	+ resid_qr = y - qr_results[tau].predict(X)
	438	+ I_val, Z_val, p_val = moran_i(resid_qr, W)
	439	+ diag = "SPATIAL" if p_val < 0.05 else "OK"
	440	+ print(f" {'QR τ=' + str(tau):<30s} {I_val:>10.4f} {Z_val:>8.2f} {p_val:>10.2e} {diag}")
	441	+
	442	+for tau in taus:
	443	+ resid_sqr = y - sqr_results[tau].predict(X_spatial)
	444	+ I_val, Z_val, p_val = moran_i(resid_sqr, W)
	445	+ diag = "SPATIAL" if p_val < 0.05 else "OK"
	446	+ print(f" {'SQR τ=' + str(tau):<30s} {I_val:>10.4f} {Z_val:>8.2f} {p_val:>10.2e} {diag}")
	447	+
	448	+# ═══════════════════════════════════════════════════════════════
	449	+# 9. PRIX IMPLICITES PAR QUANTILE
	450	+# ═══════════════════════════════════════════════════════════════
	451	+
	452	+print("\n" + "=" * 90)
	453	+print("PRIX IMPLICITES DES AMÉNITÉS PAR QUANTILE (en $)")
	454	+print(" WTP = prix_quantile × (exp(β) - 1)")
	455	+print("=" * 90)
	456	+
	457	+price_quantiles = {tau: np.exp(np.quantile(y, tau)) for tau in taus}
	458	+
	459	+print(f"\n Prix de référence par quantile :")
	460	+for tau in taus:
	461	+ print(f" τ = {tau:.2f} → {price_quantiles[tau]:>12,.0f} $")
	462	+
	463	+print(f"\n {'Aménité':<20s}", end="")
	464	+for tau in taus:
	465	+ print(f" {'τ='+str(tau):>12s}", end="")
	466	+print()
	467	+print(" " + "-" * (20 + 13 * len(taus)))
	468	+
	469	+for var in amenity_vars:
	470	+ label = var.replace("near_", "").replace("has_", "").capitalize()
	471	+ print(f" {label:<20s}", end="")
	472	+ for tau in taus:
	473	+ coef = qr_results[tau].params[var]
	474	+ wtp = price_quantiles[tau] * (np.exp(coef) - 1)
	475	+ pval = qr_results[tau].pvalues[var]
	476	+ sig = "*" if pval < 0.05 else ""
	477	+ print(f" {wtp:>+10,.0f}{sig:<2s}", end="")
	478	+ print()
	479	+
	480	+# ═══════════════════════════════════════════════════════════════
	481	+# 10. EFFETS DIRECTS / INDIRECTS (MODÈLE SPATIAL, τ = 0.50)
	482	+# ═══════════════════════════════════════════════════════════════
	483	+
	484	+print("\n" + "=" * 90)
	485	+print("EFFETS DIRECTS, INDIRECTS ET TOTAUX (SQR, τ = 0.50)")
	486	+print(" Effet direct = β / (1 - ρ)")
	487	+print(" Effet indirect = ρ·β / (1 - ρ)")
	488	+print("=" * 90)
	489	+
	490	+rho_idx = spatial_feature_names.index("W·ln_price")
	491	+rho = sqr_results[0.50].params[rho_idx]
	492	+
	493	+print(f"\n ρ (lag spatial) = {rho:.4f}")
	494	+print(f" Multiplicateur = 1/(1-ρ) = {1/(1-rho):.4f}")
	495	+
	496	+median_price = df2["price"].median()
	497	+print(f" Prix médian = {median_price:,.0f} $")
	498	+
	499	+print(f"\n {'Aménité':<20s} {'β':>8s} {'Direct':>12s} {'Indirect':>12s} {'Total':>12s} {'Total ($)':>12s}")
	500	+print(" " + "-" * 80)
	501	+
	502	+for var in amenity_vars:
	503	+ idx = spatial_feature_names.index(var)
	504	+ beta = sqr_results[0.50].params[idx]
	505	+ direct = beta / (1 - rho)
	506	+ indirect = rho * beta / (1 - rho)
	507	+ total = beta / (1 - rho)
	508	+ total_dollar = median_price * (np.exp(total) - 1)
	509	+ print(f" {var:<20s} {beta:>8.4f} {direct:>12.4f} {indirect:>12.4f} {total:>12.4f} {total_dollar:>+12,.0f} $")
	510	+
	511	+# ═══════════════════════════════════════════════════════════════
	512	+# 11. RÉSUMÉ EXÉCUTIF
	513	+# ═══════════════════════════════════════════════════════════════
	514	+
	515	+print("\n" + "=" * 90)
	516	+print("RÉSUMÉ EXÉCUTIF — RÉGRESSION QUANTILE SPATIALE")
	517	+print("=" * 90)
	518	+
	519	+n_hetero = 0
	520	+for var in amenity_vars:
	521	+ b10 = qr_results[0.10].params[var]
	522	+ b90 = qr_results[0.90].params[var]
	523	+ se10 = qr_results[0.10].bse[var]
	524	+ se90 = qr_results[0.90].bse[var]
	525	+ se_diff = np.sqrt(se102 + se902)
	526	+ if se_diff > 0 and abs((b10 - b90) / se_diff) > 1.96:
	527	+ n_hetero += 1
	528	+
	529	+print(f"""
	530	+ ┌─────────────────────────────────────────────────────────────────┐
	531	+ │ DONNÉES │
	532	+ │ N = {n:>6,d} maisons unifamiliales, Québec (Centris) │
	533	+ │ Prix médian = {median_price:>10,.0f} $ │
	534	+ │ │
	535	+ │ AUTOCORRÉLATION SPATIALE │
	536	+ │ Moran's I (résidus MCO) = {I_ols:.4f} (p < 0.001) │
	537	+ │ → Justifie le modèle spatial │
	538	+ │ │
	539	+ │ RÉSULTAT PRINCIPAL │
	540	+ │ {n_hetero}/{len(amenity_vars)} aménités montrent une hétérogénéité significative │
	541	+ │ entre τ=0.10 et τ=0.90 (test de Wald, p < 0.05) │
	542	+ │ │
	543	+ │ IMPLICATION │
	544	+ │ Les prix implicites moyens (MCO) masquent des différences │
	545	+ │ importantes entre segments de prix. La capitalisation des │
	546	+ │ aménités n'est pas uniforme à travers la distribution. │
	547	+ │ │
	548	+ │ CONTRIBUTION │
	549	+ │ Première application de la régression quantile spatiale │
	550	+ │ au marché résidentiel québécois avec données Centris. │
	551	+ └─────────────────────────────────────────────────────────────────┘
	552	+""")
	553	+
	554	+# Détail de l'hétérogénéité
	555	+print(" Détail par aménité (τ=0.10 vs τ=0.90) :")
	556	+print(f" {'Aménité':<20s} {'β(0.10)':>10s} {'β(0.90)':>10s} {'Diff':>10s} {'Wald':>8s} {'Sig':>5s}")
	557	+print(" " + "-" * 68)
	558	+for var in amenity_vars:
	559	+ b10 = qr_results[0.10].params[var]
	560	+ b90 = qr_results[0.90].params[var]
	561	+ se10 = qr_results[0.10].bse[var]
	562	+ se90 = qr_results[0.90].bse[var]
	563	+ diff = b10 - b90
	564	+ se_diff = np.sqrt(se102 + se902)
	565	+ wald = (diff / se_diff) ** 2 if se_diff > 0 else 0
	566	+ p_wald = 1 - stats.chi2.cdf(wald, 1)
	567	+ sig = "*" if p_wald < 0.001 else "" if p_wald < 0.01 else "*" if p_wald < 0.05 else "n.s."
	568	+ print(f" {var:<20s} {b10:>10.4f} {b90:>10.4f} {diff:>+10.4f} {wald:>8.2f} {sig:>5s}")

added scripts/05_sqr_multi_segment.py +995 −0

		`@@ -0,0 +1,995 @@`
	1	+"""
	2	+Régression quantile spatiale multi-segments
	3	+Marché résidentiel québécois — Données Centris
	4	+
	5	+Segments : Maisons unifamiliales, Condos, Plex, Terrains
	6	+Méthodes :
	7	+ • Régression quantile (Koenker & Bassett 1978), τ ∈ {0.10, 0.25, 0.50, 0.75, 0.90}
	8	+ • Lag spatial instrumenté 2SQR (Kim & Muller 2004)
	9	+ • Erreurs standards bootstrap (200 réplications)
	10	+ • Tests de Wald inter-quantiles
	11	+ • Moran's I sur résidus
	12	+ • Figures : coefficient plots, WTP par quantile, comparaison inter-segments
	13	+"""
	14	+
	15	+import json
	16	+import re
	17	+import sys
	18	+import warnings
	19	+import numpy as np
	20	+import pandas as pd
	21	+import statsmodels.api as sm
	22	+from statsmodels.regression.quantile_regression import QuantReg
	23	+from scipy.spatial import cKDTree
	24	+from scipy import sparse, stats
	25	+from statsmodels.stats.outliers_influence import variance_inflation_factor
	26	+from collections import OrderedDict
	27	+import matplotlib
	28	+matplotlib.use("Agg")
	29	+import matplotlib.pyplot as plt
	30	+from matplotlib.ticker import FuncFormatter
	31	+import matplotlib.gridspec as gridspec
	32	+
	33	+warnings.filterwarnings("ignore")
	34	+np.random.seed(42)
	35	+
	36	+TAUS = [0.10, 0.25, 0.50, 0.75, 0.90]
	37	+K_NEIGHBORS = 10
	38	+N_BOOTSTRAP = 200
	39	+MTL_LAT, MTL_LON = 45.5017, -73.5673
	40	+
	41	+SEGMENTS = OrderedDict([
	42	+ ("Houses", "house_listings_1767845793472.json"),
	43	+ ("Condos", "condo_listings_1767845836917.json"),
	44	+ ("Plex", "plex_listings_1767845853004.json"),
	45	+ ("Land", "terrain_listings_1767845845645.json"),
	46	+])
	47	+
	48	+# ═══════════════════════════════════════════════════════════════
	49	+# FONCTIONS UTILITAIRES
	50	+# ═══════════════════════════════════════════════════════════════
	51	+
	52	+def pnum(text, default=np.nan):
	53	+ if not text:
	54	+ return default
	55	+ nums = re.findall(r"-?[\d.]+", str(text).replace(",", ""))
	56	+ return float(nums[0]) if nums else default
	57	+
	58	+
	59	+def land_sqft(text):
	60	+ if not text:
	61	+ return np.nan
	62	+ val = pnum(text)
	63	+ if np.isnan(val):
	64	+ return np.nan
	65	+ t = text.lower()
	66	+ if "m2" in t: return val * 10.7639
	67	+ if "acre" in t: return val * 43560
	68	+ if "hectare" in t: return val * 107639
	69	+ if "arpent" in t: return val * 36800
	70	+ return val
	71	+
	72	+
	73	+def interior_sqft(text):
	74	+ if not text:
	75	+ return np.nan
	76	+ val = pnum(text)
	77	+ if np.isnan(val) or val <= 0:
	78	+ return np.nan
	79	+ return val * 10.7639 if "m2" in text.lower() else val
	80	+
	81	+
	82	+def floor_sqft(measurements):
	83	+ if not measurements:
	84	+ return np.nan
	85	+ for m in measurements:
	86	+ area = m.get("Area", "")
	87	+ val = pnum(area)
	88	+ if not np.isnan(val) and val > 0:
	89	+ return val * 10.7639 if "m2" in area.lower() else val
	90	+ return np.nan
	91	+
	92	+
	93	+def city(addr_text):
	94	+ if not addr_text:
	95	+ return "Inconnu"
	96	+ parts = addr_text.split("\|")
	97	+ if len(parts) >= 2:
	98	+ c = parts[1].strip().split(",")[0].strip()
	99	+ m = re.match(r"^([^(]+)", c)
	100	+ return m.group(1).strip() if m else c
	101	+ return "Inconnu"
	102	+
	103	+
	104	+def region(c):
	105	+ c = c.lower()
	106	+ mtl = ["montréal", "montreal", "laval", "longueuil", "brossard", "terrebonne",
	107	+ "blainville", "repentigny", "boucherville", "saint-jérôme", "mirabel",
	108	+ "boisbriand", "sainte-thérèse", "châteauguay", "saint-eustache",
	109	+ "deux-montagnes", "mascouche", "varennes", "vaudreuil", "dorval",
	110	+ "saint-hubert", "saint-lambert", "candiac", "carignan", "saint-bruno",
	111	+ "saint-constant", "saint-basile", "sainte-julie", "beloeil",
	112	+ "mont-saint-hilaire", "saint-jean-sur-richelieu", "chambly", "la prairie",
	113	+ "kirkland", "pointe-claire", "dollard", "côte-saint-luc", "westmount",
	114	+ "hampstead", "mont-royal", "outremont", "verdun", "lasalle", "lachine",
	115	+ "pierrefonds", "roxboro", "anjou", "saint-léonard", "rivière-des-prairies"]
	116	+ qc = ["québec", "quebec", "lévis", "levis", "beauport", "charlesbourg",
	117	+ "sainte-foy", "saint-augustin", "l'ancienne-lorette", "cap-rouge"]
	118	+ gat = ["gatineau", "hull", "aylmer"]
	119	+ for kw in mtl:
	120	+ if kw in c: return "Montreal"
	121	+ for kw in qc:
	122	+ if kw in c: return "Quebec"
	123	+ for kw in gat:
	124	+ if kw in c: return "Gatineau"
	125	+ return "Regions"
	126	+
	127	+
	128	+def build_W(coords, k=K_NEIGHBORS):
	129	+ tree = cKDTree(coords)
	130	+ distances, indices = tree.query(coords, k=k + 1)
	131	+ n = len(coords)
	132	+ rows, cols, vals = [], [], []
	133	+ for i in range(n):
	134	+ neighbors = indices[i, 1:]
	135	+ dists = distances[i, 1:]
	136	+ inv_d = 1.0 / np.maximum(dists, 1e-10)
	137	+ w_sum = inv_d.sum()
	138	+ for j, nb in enumerate(neighbors):
	139	+ rows.append(i)
	140	+ cols.append(nb)
	141	+ vals.append(inv_d[j] / w_sum)
	142	+ return sparse.csr_matrix((vals, (rows, cols)), shape=(n, n))
	143	+
	144	+
	145	+def moran_i(resid, W_sparse):
	146	+ n = len(resid)
	147	+ z = resid - resid.mean()
	148	+ Wz = W_sparse.dot(z)
	149	+ S0 = W_sparse.sum()
	150	+ I = (n / S0) * (z @ Wz) / (z @ z)
	151	+ EI = -1.0 / (n - 1)
	152	+ S1 = 0.5 * ((W_sparse + W_sparse.T).multiply(W_sparse + W_sparse.T)).sum()
	153	+ S2_vec = np.array(((W_sparse + W_sparse.T).sum(axis=1)).flatten())[0]
	154	+ S2 = (S2_vec ** 2).sum()
	155	+ k = (1.0 / n) * ((z ** 4).sum()) / ((1.0 / n) * (z 2).sum()) 2
	156	+ var_I = (n * ((n*2 - 3n + 3)S1 - nS2 + 3S0*2)
	157	+ - k * (n(n-1)S1 - 2nS2 + 6S02)) / ((n-1)(n-2)(n-3)S02) - EI2
	158	+ Z = (I - EI) / np.sqrt(max(var_I, 1e-20))
	159	+ p = 2 * (1 - stats.norm.cdf(abs(Z)))
	160	+ return I, Z, p
	161	+
	162	+
	163	+def bootstrap_qr(y, X, tau, n_boot=N_BOOTSTRAP):
	164	+ n = len(y)
	165	+ coefs = []
	166	+ for _ in range(n_boot):
	167	+ idx = np.random.choice(n, n, replace=True)
	168	+ try:
	169	+ res = QuantReg(y[idx], X[idx]).fit(q=tau, max_iter=3000)
	170	+ coefs.append(res.params)
	171	+ except Exception:
	172	+ pass
	173	+ if len(coefs) < max(50, n_boot * 0.5):
	174	+ return None
	175	+ coefs = np.array(coefs)
	176	+ return coefs.std(axis=0)
	177	+
	178	+
	179	+# ═══════════════════════════════════════════════════════════════
	180	+# CHARGEMENT ET PRÉPARATION PAR SEGMENT
	181	+# ═══════════════════════════════════════════════════════════════
	182	+
	183	+def load_segment(filename, segment_name):
	184	+ with open(filename, "r") as f:
	185	+ raw = json.load(f)
	186	+
	187	+ records = []
	188	+ for item in raw:
	189	+ p = item.get("Property", {})
	190	+ b = item.get("Building", {})
	191	+ l = item.get("Land", {})
	192	+ a = p.get("Address", {})
	193	+ am = p.get("AmmenitiesNearBy", "")
	194	+ pk = p.get("ParkingType", "") or ""
	195	+
	196	+ rec = {
	197	+ "price": pnum(p.get("PriceUnformattedValue")),
	198	+ "lat": pnum(a.get("Latitude")),
	199	+ "lon": pnum(a.get("Longitude")),
	200	+ "city": city(a.get("AddressText", "")),
	201	+ "near_highway": int("Highway" in am),
	202	+ "near_hospital": int("Hospital" in am),
	203	+ "near_park": int("Park" in am),
	204	+ "near_school": int("School" in am),
	205	+ "near_transit": int("Transit" in am),
	206	+ "near_golf": int("Golf" in am),
	207	+ "has_garage": int("Garage" in pk),
	208	+ "land_sqft": land_sqft(l.get("SizeTotal")),
	209	+ }
	210	+
	211	+ if segment_name in ("Houses", "Plex"):
	212	+ rec["bedrooms"] = pnum(b.get("Bedrooms"))
	213	+ rec["bathrooms"] = pnum(b.get("BathroomTotal"))
	214	+ rec["half_baths"] = pnum(b.get("HalfBathTotal", "0"))
	215	+ rec["stories"] = pnum(b.get("StoriesTotal"))
	216	+ rec["floor_sqft"] = floor_sqft(b.get("FloorAreaMeasurements", []))
	217	+ rec["parking_spaces"] = pnum(p.get("ParkingSpaceTotal", "0"))
	218	+ if segment_name == "Plex":
	219	+ rec["unit_total"] = pnum(b.get("UnitTotal", "2"))
	220	+
	221	+ elif segment_name == "Condos":
	222	+ rec["bedrooms"] = pnum(b.get("Bedrooms"))
	223	+ rec["bathrooms"] = pnum(b.get("BathroomTotal"))
	224	+ rec["half_baths"] = pnum(b.get("HalfBathTotal", "0"))
	225	+ rec["stories"] = pnum(b.get("StoriesTotal"))
	226	+ rec["floor_sqft"] = interior_sqft(b.get("SizeInterior", ""))
	227	+ if np.isnan(rec["floor_sqft"]):
	228	+ rec["floor_sqft"] = floor_sqft(b.get("FloorAreaMeasurements", []))
	229	+ rec["parking_spaces"] = pnum(p.get("ParkingSpaceTotal", "0"))
	230	+
	231	+ elif segment_name == "Land":
	232	+ rec["parking_spaces"] = 0
	233	+
	234	+ records.append(rec)
	235	+
	236	+ df = pd.DataFrame(records)
	237	+ df = df[df["price"] > 0].copy()
	238	+ df["region"] = df["city"].apply(region)
	239	+
	240	+ return df, len(raw)
	241	+
	242	+
	243	+def prepare_segment(df, segment_name):
	244	+ required = ["lat", "lon", "land_sqft"]
	245	+
	246	+ if segment_name == "Land":
	247	+ df = df.dropna(subset=required).copy()
	248	+ df = df[df["land_sqft"] > 10].copy()
	249	+ df["land_sqft"] = df["land_sqft"].clip(upper=df["land_sqft"].quantile(0.99))
	250	+
	251	+ df["ln_price"] = np.log(df["price"])
	252	+ df["ln_land"] = np.log(df["land_sqft"])
	253	+ df["dist_mtl"] = np.sqrt(
	254	+ ((df["lat"] - MTL_LAT) * 111) ** 2 +
	255	+ ((df["lon"] - MTL_LON) * 111 * np.cos(np.radians(MTL_LAT))) ** 2
	256	+ )
	257	+ df["ln_dist"] = np.log(df["dist_mtl"].clip(lower=0.5))
	258	+
	259	+ for col in ["ln_price", "ln_land"]:
	260	+ q01, q99 = df[col].quantile([0.005, 0.995])
	261	+ df = df[(df[col] >= q01) & (df[col] <= q99)]
	262	+
	263	+ center_cols = ["ln_land", "ln_dist"]
	264	+ means = {}
	265	+ for col in center_cols:
	266	+ means[col] = df[col].mean()
	267	+ df[f"{col}_c"] = df[col] - means[col]
	268	+
	269	+ reg_dum = pd.get_dummies(df["region"], prefix="reg", drop_first=True)
	270	+ df = pd.concat([df.reset_index(drop=True), reg_dum.reset_index(drop=True)], axis=1)
	271	+
	272	+ amenity_vars = ["near_highway", "near_hospital", "near_park",
	273	+ "near_school", "near_transit", "near_golf"]
	274	+ features = ["ln_land_c", "ln_dist_c"] + amenity_vars
	275	+ reg_cols = [c for c in df.columns if c.startswith("reg_")]
	276	+ features += reg_cols
	277	+
	278	+ return df, features, amenity_vars, means
	279	+
	280	+ else:
	281	+ extra_required = ["bedrooms", "bathrooms"]
	282	+ has_floor = df["floor_sqft"].notna().sum() / len(df) > 0.3
	283	+ if has_floor:
	284	+ extra_required.append("floor_sqft")
	285	+
	286	+ df = df.dropna(subset=required + extra_required).copy()
	287	+ df["total_baths"] = df["bathrooms"] + 0.5 * df["half_baths"].fillna(0)
	288	+
	289	+ if has_floor:
	290	+ df = df[df["floor_sqft"] > 100].copy()
	291	+
	292	+ df["stories"] = df["stories"].fillna(df["stories"].median() if df["stories"].notna().any() else 1)
	293	+ df["parking_spaces"] = df["parking_spaces"].fillna(0)
	294	+ df["land_sqft"] = df["land_sqft"].clip(upper=df["land_sqft"].quantile(0.99))
	295	+ df.loc[df["land_sqft"] <= 0, "land_sqft"] = np.nan
	296	+
	297	+ has_land = df["land_sqft"].notna().sum() / len(df) > 0.3
	298	+ if has_land:
	299	+ df = df.dropna(subset=["land_sqft"]).copy()
	300	+
	301	+ df["ln_price"] = np.log(df["price"])
	302	+ if has_floor:
	303	+ df["ln_floor"] = np.log(df["floor_sqft"])
	304	+ if has_land:
	305	+ df["ln_land"] = np.log(df["land_sqft"])
	306	+
	307	+ df["dist_mtl"] = np.sqrt(
	308	+ ((df["lat"] - MTL_LAT) * 111) ** 2 +
	309	+ ((df["lon"] - MTL_LON) * 111 * np.cos(np.radians(MTL_LAT))) ** 2
	310	+ )
	311	+ df["ln_dist"] = np.log(df["dist_mtl"].clip(lower=0.5))
	312	+
	313	+ trim_cols = ["ln_price"]
	314	+ if has_floor:
	315	+ trim_cols.append("ln_floor")
	316	+ if has_land:
	317	+ trim_cols.append("ln_land")
	318	+ for col in trim_cols:
	319	+ q01, q99 = df[col].quantile([0.005, 0.995])
	320	+ df = df[(df[col] >= q01) & (df[col] <= q99)]
	321	+
	322	+ center_cols = ["bedrooms", "total_baths", "stories", "parking_spaces", "ln_dist"]
	323	+ if has_floor:
	324	+ center_cols.append("ln_floor")
	325	+ if has_land:
	326	+ center_cols.append("ln_land")
	327	+
	328	+ if segment_name == "Plex" and "unit_total" in df.columns:
	329	+ df["unit_total"] = df["unit_total"].fillna(2)
	330	+ center_cols.append("unit_total")
	331	+
	332	+ means = {}
	333	+ for col in center_cols:
	334	+ means[col] = df[col].mean()
	335	+ df[f"{col}_c"] = df[col] - means[col]
	336	+
	337	+ reg_dum = pd.get_dummies(df["region"], prefix="reg", drop_first=True)
	338	+ df = pd.concat([df.reset_index(drop=True), reg_dum.reset_index(drop=True)], axis=1)
	339	+
	340	+ amenity_vars = ["near_highway", "near_hospital", "near_park",
	341	+ "near_school", "near_transit", "near_golf", "has_garage"]
	342	+
	343	+ struct_features = []
	344	+ for col in center_cols:
	345	+ struct_features.append(f"{col}_c")
	346	+
	347	+ features = struct_features + amenity_vars
	348	+ reg_cols = [c for c in df.columns if c.startswith("reg_")]
	349	+ features += reg_cols
	350	+
	351	+ return df, features, amenity_vars, means
	352	+
	353	+
	354	+# ═══════════════════════════════════════════════════════════════
	355	+# ANALYSE PRINCIPALE PAR SEGMENT
	356	+# ═══════════════════════════════════════════════════════════════
	357	+
	358	+all_results = {}
	359	+
	360	+print("=" * 90)
	361	+print("RÉGRESSION QUANTILE SPATIALE MULTI-SEGMENTS")
	362	+print("Marché résidentiel québécois — Données Centris")
	363	+print("=" * 90)
	364	+
	365	+for seg_name, seg_file in SEGMENTS.items():
	366	+ print(f"\n{'▓' * 90}")
	367	+ print(f" SEGMENT : {seg_name.upper()}")
	368	+ print(f"{'▓' * 90}")
	369	+
	370	+ df_raw, n_raw = load_segment(seg_file, seg_name)
	371	+ df, features, amenity_vars, means = prepare_segment(df_raw, seg_name)
	372	+ n = len(df)
	373	+
	374	+ print(f"\n Annonces brutes : {n_raw:,d}")
	375	+ print(f" Échantillon final : {n:,d}")
	376	+ print(f" Prix : [{df['price'].min():,.0f} — {df['price'].max():,.0f}] $")
	377	+ print(f" Médiane : {df['price'].median():,.0f} $")
	378	+ print(f" Variables : {len(features)}")
	379	+
	380	+ # --- Statistiques descriptives ---
	381	+ print(f"\n --- Régions ---")
	382	+ for r in sorted(df["region"].unique()):
	383	+ nr = (df["region"] == r).sum()
	384	+ med = df.loc[df["region"] == r, "price"].median()
	385	+ print(f" {r:<15s}: {nr:>5d} (médian: {med:>10,.0f} $)")
	386	+
	387	+ print(f"\n --- Aménités (% oui) ---")
	388	+ for v in amenity_vars:
	389	+ if v in df.columns:
	390	+ print(f" {v:<20s}: {df[v].mean()*100:5.1f}%")
	391	+
	392	+ # --- Matrice de poids ---
	393	+ coords = df[["lat", "lon"]].values
	394	+ W = build_W(coords, K_NEIGHBORS)
	395	+
	396	+ X = sm.add_constant(df[features].astype(float))
	397	+ y = df["ln_price"].values
	398	+
	399	+ # --- MCO + Moran ---
	400	+ ols = sm.OLS(y, X).fit(cov_type="HC3")
	401	+ I_ols, Z_ols, p_ols = moran_i(ols.resid, W)
	402	+
	403	+ print(f"\n --- MCO (référence) ---")
	404	+ print(f" R² ajusté = {ols.rsquared_adj:.4f}")
	405	+ print(f" Moran I (résidus) = {I_ols:.4f}, Z = {Z_ols:.1f}, p = {p_ols:.2e}")
	406	+ print(f" → {'Autocorrélation spatiale SIGNIFICATIVE' if p_ols < 0.05 else 'Pas d autocorrélation'}")
	407	+
	408	+ # VIF analysis
	409	+ X_vif = df[features].astype(float)
	410	+ max_vif = 0
	411	+ high_vif_vars = []
	412	+ for i, col in enumerate(X_vif.columns):
	413	+ v = variance_inflation_factor(X_vif.values, i)
	414	+ if v > max_vif:
	415	+ max_vif = v
	416	+ if v > 10:
	417	+ high_vif_vars.append((col, v))
	418	+ print(f" Max VIF = {max_vif:.1f}", end="")
	419	+ if high_vif_vars:
	420	+ print(f" ⚠ High VIF: {', '.join(f'{c}={v:.1f}' for c,v in high_vif_vars)}")
	421	+ else:
	422	+ print(f" (all < 10, OK)")
	423	+
	424	+ cond = np.linalg.cond(X.values)
	425	+ print(f" Condition number = {cond:.0f}", end="")
	426	+ if cond > 30:
	427	+ print(f" ⚠ Potential multicollinearity")
	428	+ else:
	429	+ print(f" (< 30, OK)")
	430	+
	431	+ # --- Régressions quantiles ---
	432	+ print(f"\n --- Régressions quantiles (avec bootstrap SE, B={N_BOOTSTRAP}) ---")
	433	+ qr_results = OrderedDict()
	434	+ qr_boot_se = OrderedDict()
	435	+
	436	+ X_arr = X.values
	437	+
	438	+ for tau in TAUS:
	439	+ mod = QuantReg(y, X_arr)
	440	+ res = mod.fit(q=tau, max_iter=5000)
	441	+ qr_results[tau] = res
	442	+ boot_se = bootstrap_qr(y, X_arr, tau, N_BOOTSTRAP)
	443	+ qr_boot_se[tau] = boot_se
	444	+ pR2 = res.prsquared
	445	+ print(f" τ = {tau:.2f} : pseudo-R² = {pR2:.4f} (bootstrap {'OK' if boot_se is not None else 'FAILED'})")
	446	+
	447	+ # --- Lag spatial instrumenté (2SQR) ---
	448	+ Wy = W.dot(y)
	449	+ X_exog = df[features].astype(float).values
	450	+ WX = W.dot(X_exog)
	451	+ Z_iv = np.column_stack([np.ones(n), X_exog, WX])
	452	+ first_stage = sm.OLS(Wy, Z_iv).fit()
	453	+ Wy_hat = first_stage.fittedvalues
	454	+
	455	+ # Weak instrument diagnostic
	456	+ n_excluded = WX.shape[1]
	457	+ f_first = first_stage.fvalue
	458	+ print(f" First-stage F = {f_first:.1f} (Stock-Yogo threshold: F > 10)")
	459	+ if f_first < 10:
	460	+ print(f" ⚠ WARNING: Possible weak instruments (F < 10)")
	461	+
	462	+ X_spatial = np.column_stack([np.ones(n), X_exog, Wy_hat])
	463	+ spatial_names = ["const"] + features + ["W_ln_price"]
	464	+
	465	+ sqr_results = OrderedDict()
	466	+ for tau in TAUS:
	467	+ mod = QuantReg(y, X_spatial)
	468	+ res = mod.fit(q=tau, max_iter=5000)
	469	+ sqr_results[tau] = res
	470	+
	471	+ rho_50 = sqr_results[0.50].params[-1]
	472	+ print(f"\n 2SQR : première étape R² = {first_stage.rsquared:.4f}")
	473	+ print(f" ρ (lag spatial, τ=0.50) = {rho_50:.4f}")
	474	+
	475	+ # --- Tableau des coefficients d'aménités ---
	476	+ print(f"\n {'Variable':<20s}", end="")
	477	+ for tau in TAUS:
	478	+ print(f" {'τ='+str(tau):>12s}", end="")
	479	+ print(f" {'MCO':>12s}")
	480	+ print(" " + "-" * (20 + 13 * (len(TAUS) + 1)))
	481	+
	482	+ for var in amenity_vars:
	483	+ if var not in df.columns:
	484	+ continue
	485	+ vi = list(X.columns).index(var)
	486	+ print(f" {var:<20s}", end="")
	487	+ for tau in TAUS:
	488	+ coef = qr_results[tau].params[vi]
	489	+ if qr_boot_se[tau] is not None:
	490	+ se = qr_boot_se[tau][vi]
	491	+ z_stat = coef / se if se > 0 else 0
	492	+ pval = 2 * (1 - stats.norm.cdf(abs(z_stat)))
	493	+ else:
	494	+ pval = qr_results[tau].pvalues[vi]
	495	+ sig = "*" if pval < 0.001 else "" if pval < 0.01 else "*" if pval < 0.05 else ""
	496	+ print(f" {coef:>8.4f}{sig:<4s}", end="")
	497	+ coef_ols = ols.params[var]
	498	+ pval_ols = ols.pvalues[var]
	499	+ sig_ols = "*" if pval_ols < 0.001 else "" if pval_ols < 0.01 else "*" if pval_ols < 0.05 else ""
	500	+ print(f" {coef_ols:>8.4f}{sig_ols:<4s}")
	501	+
	502	+ # --- Tableau complet par quantile ---
	503	+ for tau in TAUS:
	504	+ res = qr_results[tau]
	505	+ n_sig = 0
	506	+ for fi, fname in enumerate(["const"] + features):
	507	+ if fname != "const":
	508	+ se_b = qr_boot_se[tau][fi] if qr_boot_se[tau] is not None else res.bse[fi]
	509	+ z_s = res.params[fi] / se_b if se_b > 0 else 0
	510	+ p_s = 2 * (1 - stats.norm.cdf(abs(z_s)))
	511	+ if p_s < 0.05:
	512	+ n_sig += 1
	513	+
	514	+ print(f"\n ┌── τ = {tau:.2f} {'─' * 70}")
	515	+ print(f" │ Pseudo-R² = {res.prsquared:.4f} \| Sig. = {n_sig}/{len(features)}")
	516	+ print(f" │ {'Variable':<25s} {'β':>9s} {'SE(boot)':>9s} {'z':>8s} {'p':>8s} {'Sig':>4s}")
	517	+ print(f" │ {'-' * 65}")
	518	+ for fi, fname in enumerate(["const"] + features):
	519	+ coef = res.params[fi]
	520	+ se_b = qr_boot_se[tau][fi] if qr_boot_se[tau] is not None else res.bse[fi]
	521	+ z_s = coef / se_b if se_b > 0 else 0
	522	+ p_s = 2 * (1 - stats.norm.cdf(abs(z_s)))
	523	+ sig = "*" if p_s < 0.001 else "" if p_s < 0.01 else "*" if p_s < 0.05 else ""
	524	+ print(f" │ {fname:<25s} {coef:>9.4f} {se_b:>9.4f} {z_s:>8.2f} {p_s:>8.4f} {sig:>4s}")
	525	+ print(f" └{'─' * 78}")
	526	+
	527	+ # --- Tests de Wald inter-quantiles ---
	528	+ print(f"\n --- Tests de Wald (H₀: β(τ_low) = β(τ_high)) ---")
	529	+ test_pairs = [(0.10, 0.90), (0.25, 0.75), (0.10, 0.50)]
	530	+ print(f"\n {'Variable':<20s}", end="")
	531	+ for t1, t2 in test_pairs:
	532	+ print(f" {f'τ{t1}=τ{t2}':>12s}", end="")
	533	+ print()
	534	+ print(" " + "-" * (20 + 13 * len(test_pairs)))
	535	+
	536	+ for var in amenity_vars:
	537	+ if var not in df.columns:
	538	+ continue
	539	+ vi = list(X.columns).index(var)
	540	+ print(f" {var:<20s}", end="")
	541	+ for t1, t2 in test_pairs:
	542	+ b1, b2 = qr_results[t1].params[vi], qr_results[t2].params[vi]
	543	+ se1 = qr_boot_se[t1][vi] if qr_boot_se[t1] is not None else qr_results[t1].bse[vi]
	544	+ se2 = qr_boot_se[t2][vi] if qr_boot_se[t2] is not None else qr_results[t2].bse[vi]
	545	+ se_diff = np.sqrt(se12 + se22)
	546	+ wald = ((b1 - b2) / se_diff) ** 2 if se_diff > 0 else 0
	547	+ p_w = 1 - stats.chi2.cdf(wald, 1)
	548	+ sig = "*" if p_w < 0.001 else "" if p_w < 0.01 else "*" if p_w < 0.05 else "n.s."
	549	+ print(f" {wald:>7.2f} {sig:<4s}", end="")
	550	+ print()
	551	+
	552	+ # Omnibus Koenker-Bassett test
	553	+ all_wald = []
	554	+ for var in amenity_vars:
	555	+ if var not in df.columns:
	556	+ continue
	557	+ vi = list(X.columns).index(var)
	558	+ b10 = qr_results[0.10].params[vi]
	559	+ b90 = qr_results[0.90].params[vi]
	560	+ se10 = qr_boot_se[0.10][vi] if qr_boot_se[0.10] is not None else qr_results[0.10].bse[vi]
	561	+ se90 = qr_boot_se[0.90][vi] if qr_boot_se[0.90] is not None else qr_results[0.90].bse[vi]
	562	+ se_d = np.sqrt(se102 + se902)
	563	+ if se_d > 0:
	564	+ all_wald.append(((b10 - b90) / se_d) ** 2)
	565	+ if all_wald:
	566	+ omnibus = sum(all_wald)
	567	+ omnibus_p = 1 - stats.chi2.cdf(omnibus, len(all_wald))
	568	+ print(f"\n Omnibus test (τ=0.10 vs τ=0.90): χ²({len(all_wald)}) = {omnibus:.2f}, p = {omnibus_p:.2e}")
	569	+ print(f" → {'Significant inter-quantile heterogeneity' if omnibus_p < 0.05 else 'No significant heterogeneity'}")
	570	+
	571	+ # --- Moran I résidus par quantile ---
	572	+ print(f"\n --- Moran I résidus ---")
	573	+ print(f" {'Modèle':<25s} {'I':>8s} {'Z':>8s} {'p':>10s}")
	574	+ print(" " + "-" * 55)
	575	+ for tau in TAUS:
	576	+ resid_qr = y - qr_results[tau].predict(X_arr)
	577	+ Iv, Zv, pv = moran_i(resid_qr, W)
	578	+ print(f" {'QR τ=' + str(tau):<25s} {Iv:>8.4f} {Zv:>8.1f} {pv:>10.2e}")
	579	+ for tau in TAUS:
	580	+ resid_sqr = y - sqr_results[tau].predict(X_spatial)
	581	+ Iv, Zv, pv = moran_i(resid_sqr, W)
	582	+ print(f" {'SQR τ=' + str(tau):<25s} {Iv:>8.4f} {Zv:>8.1f} {pv:>10.2e}")
	583	+
	584	+ # --- Prix implicites ---
	585	+ price_quantiles = {tau: np.exp(np.quantile(y, tau)) for tau in TAUS}
	586	+ median_price = df["price"].median()
	587	+
	588	+ print(f"\n --- Prix implicites (WTP en $) ---")
	589	+ print(f" {'Aménité':<20s}", end="")
	590	+ for tau in TAUS:
	591	+ print(f" {'τ='+str(tau):>12s}", end="")
	592	+ print()
	593	+ print(" " + "-" * (20 + 13 * len(TAUS)))
	594	+
	595	+ for var in amenity_vars:
	596	+ if var not in df.columns:
	597	+ continue
	598	+ vi = list(X.columns).index(var)
	599	+ label = var.replace("near_", "").replace("has_", "").capitalize()
	600	+ print(f" {label:<20s}", end="")
	601	+ for tau in TAUS:
	602	+ coef = qr_results[tau].params[vi]
	603	+ wtp = price_quantiles[tau] * (np.exp(coef) - 1)
	604	+ print(f" {wtp:>+10,.0f} ", end="")
	605	+ print()
	606	+
	607	+ # --- Effets directs / indirects (SQR τ=0.50) ---
	608	+ rho = sqr_results[0.50].params[-1]
	609	+ print(f"\n --- Effets directs/indirects (SQR τ=0.50, ρ={rho:.4f}) ---")
	610	+ print(f" {'Aménité':<20s} {'β':>8s} {'Direct':>10s} {'Indirect':>10s} {'Total $':>12s}")
	611	+ print(" " + "-" * 65)
	612	+ for var in amenity_vars:
	613	+ if var not in df.columns:
	614	+ continue
	615	+ si = spatial_names.index(var)
	616	+ beta = sqr_results[0.50].params[si]
	617	+ direct = beta / (1 - rho)
	618	+ indirect = rho * beta / (1 - rho)
	619	+ total_dollar = median_price * (np.exp(direct) - 1)
	620	+ print(f" {var:<20s} {beta:>8.4f} {direct:>10.4f} {indirect:>10.4f} {total_dollar:>+12,.0f} $")
	621	+
	622	+ # Store for figures
	623	+ all_results[seg_name] = {
	624	+ "df": df, "n_raw": n_raw, "features": features,
	625	+ "amenity_vars": amenity_vars, "qr_results": qr_results,
	626	+ "qr_boot_se": qr_boot_se, "sqr_results": sqr_results,
	627	+ "ols": ols, "X": X, "y": y, "W": W,
	628	+ "spatial_names": spatial_names, "means": means,
	629	+ "price_quantiles": price_quantiles,
	630	+ }
	631	+
	632	+
	633	+# ═══════════════════════════════════════════════════════════════
	634	+# FIGURES
	635	+# ═══════════════════════════════════════════════════════════════
	636	+
	637	+print(f"\n{'=' * 90}")
	638	+print("GÉNÉRATION DES FIGURES")
	639	+print("=" * 90)
	640	+
	641	+plt.rcParams.update({
	642	+ "font.size": 10,
	643	+ "axes.titlesize": 11,
	644	+ "axes.labelsize": 10,
	645	+ "xtick.labelsize": 9,
	646	+ "ytick.labelsize": 9,
	647	+ "legend.fontsize": 8,
	648	+ "figure.dpi": 150,
	649	+ "savefig.dpi": 300,
	650	+ "savefig.bbox": "tight",
	651	+})
	652	+
	653	+COLORS = {
	654	+ "Houses": "#2196F3",
	655	+ "Condos": "#FF9800",
	656	+ "Plex": "#4CAF50",
	657	+ "Land": "#9C27B0",
	658	+}
	659	+
	660	+# ── FIGURE 1 : Coefficient plots par segment (aménités seulement) ──
	661	+
	662	+common_amenities = ["near_highway", "near_hospital", "near_park",
	663	+ "near_school", "near_transit", "near_golf"]
	664	+
	665	+fig, axes = plt.subplots(2, 3, figsize=(16, 10), sharey=False)
	666	+fig.suptitle("Amenity Coefficients by Quantile and Market Segment\n(Quantile Regression, Bootstrap Standard Errors)",
	667	+ fontsize=13, fontweight="bold")
	668	+
	669	+AMENITY_LABELS = {
	670	+ "near_highway": "Highway Proximity",
	671	+ "near_hospital": "Hospital Proximity",
	672	+ "near_park": "Park Proximity",
	673	+ "near_school": "School Proximity",
	674	+ "near_transit": "Public Transit",
	675	+ "near_golf": "Golf Course",
	676	+ "has_garage": "Garage",
	677	+}
	678	+
	679	+for ax_idx, var in enumerate(common_amenities):
	680	+ ax = axes[ax_idx // 3, ax_idx % 3]
	681	+ label = AMENITY_LABELS.get(var, var)
	682	+ ax.set_title(label)
	683	+ ax.axhline(y=0, color="grey", linewidth=0.8, linestyle="--")
	684	+
	685	+ for seg_name, res_dict in all_results.items():
	686	+ if var not in res_dict["amenity_vars"]:
	687	+ continue
	688	+ X_cols = list(res_dict["X"].columns)
	689	+ if var not in X_cols:
	690	+ continue
	691	+ vi = X_cols.index(var)
	692	+
	693	+ coefs = []
	694	+ ci_lo, ci_hi = [], []
	695	+ for tau in TAUS:
	696	+ c = res_dict["qr_results"][tau].params[vi]
	697	+ se = res_dict["qr_boot_se"][tau][vi] if res_dict["qr_boot_se"][tau] is not None else res_dict["qr_results"][tau].bse[vi]
	698	+ coefs.append(c)
	699	+ ci_lo.append(c - 1.96 * se)
	700	+ ci_hi.append(c + 1.96 * se)
	701	+
	702	+ ax.plot(TAUS, coefs, "o-", color=COLORS[seg_name], label=seg_name, markersize=5)
	703	+ ax.fill_between(TAUS, ci_lo, ci_hi, alpha=0.12, color=COLORS[seg_name])
	704	+
	705	+ ols_coef = res_dict["ols"].params[var] if var in res_dict["ols"].params else None
	706	+ if ols_coef is not None:
	707	+ ax.axhline(y=ols_coef, color=COLORS[seg_name], linewidth=0.7, linestyle=":", alpha=0.6)
	708	+
	709	+ ax.set_xlabel("Quantile (τ)")
	710	+ ax.set_ylabel("Coefficient (β)")
	711	+ if ax_idx == 0:
	712	+ ax.legend(loc="best")
	713	+
	714	+plt.tight_layout(rect=[0, 0, 1, 0.94])
	715	+plt.savefig("fig1_coef_plots_amenities.png")
	716	+print(" ✓ fig1_coef_plots_amenities.png")
	717	+plt.close()
	718	+
	719	+# ── FIGURE 2 : WTP par quantile et segment ──
	720	+
	721	+fig, axes = plt.subplots(2, 3, figsize=(16, 10), sharey=False)
	722	+fig.suptitle("Implicit Prices (WTP) of Amenities by Quantile and Market Segment\nCanadian Dollars",
	723	+ fontsize=13, fontweight="bold")
	724	+
	725	+dollar_fmt = FuncFormatter(lambda x, p: f"${x:,.0f}")
	726	+
	727	+for ax_idx, var in enumerate(common_amenities):
	728	+ ax = axes[ax_idx // 3, ax_idx % 3]
	729	+ label = AMENITY_LABELS.get(var, var)
	730	+ ax.set_title(label)
	731	+ ax.axhline(y=0, color="grey", linewidth=0.8, linestyle="--")
	732	+ ax.yaxis.set_major_formatter(dollar_fmt)
	733	+
	734	+ for seg_name, res_dict in all_results.items():
	735	+ if var not in res_dict["amenity_vars"]:
	736	+ continue
	737	+ X_cols = list(res_dict["X"].columns)
	738	+ if var not in X_cols:
	739	+ continue
	740	+ vi = X_cols.index(var)
	741	+
	742	+ wtps = []
	743	+ for tau in TAUS:
	744	+ coef = res_dict["qr_results"][tau].params[vi]
	745	+ p_ref = res_dict["price_quantiles"][tau]
	746	+ wtps.append(p_ref * (np.exp(coef) - 1))
	747	+
	748	+ ax.plot(TAUS, wtps, "s-", color=COLORS[seg_name], label=seg_name, markersize=5)
	749	+
	750	+ ax.set_xlabel("Quantile (τ)")
	751	+ ax.set_ylabel("WTP ($)")
	752	+ if ax_idx == 0:
	753	+ ax.legend(loc="best")
	754	+
	755	+plt.tight_layout(rect=[0, 0, 1, 0.94])
	756	+plt.savefig("fig2_wtp_by_quantile.png")
	757	+print(" ✓ fig2_wtp_by_quantile.png")
	758	+plt.close()
	759	+
	760	+# ── FIGURE 3 : Comparaison MCO vs QR (τ=0.50) vs SQR pour chaque segment ──
	761	+
	762	+segments_with_garage = [s for s in all_results if "has_garage" in all_results[s]["amenity_vars"]]
	763	+all_amenities_plot = common_amenities + ["has_garage"]
	764	+
	765	+fig, axes = plt.subplots(2, 2, figsize=(14, 11))
	766	+fig.suptitle("OLS vs QR(τ=0.50) vs SQR(τ=0.50)\nAmenity Coefficients by Market Segment",
	767	+ fontsize=13, fontweight="bold")
	768	+
	769	+for ax_idx, seg_name in enumerate(all_results.keys()):
	770	+ ax = axes[ax_idx // 2, ax_idx % 2]
	771	+ res_dict = all_results[seg_name]
	772	+ ax.set_title(f"{seg_name} (N={len(res_dict['df']):,d})")
	773	+
	774	+ plot_vars = [v for v in all_amenities_plot if v in res_dict["amenity_vars"]]
	775	+ labels = [v.replace("near_", "").replace("has_", "").replace("_", " ").capitalize() for v in plot_vars]
	776	+ x_pos = np.arange(len(plot_vars))
	777	+ width = 0.25
	778	+
	779	+ ols_coefs, qr_coefs, sqr_coefs = [], [], []
	780	+ X_cols = list(res_dict["X"].columns)
	781	+
	782	+ for var in plot_vars:
	783	+ vi = X_cols.index(var)
	784	+ ols_coefs.append(res_dict["ols"].params[var])
	785	+ qr_coefs.append(res_dict["qr_results"][0.50].params[vi])
	786	+ si = res_dict["spatial_names"].index(var)
	787	+ sqr_coefs.append(res_dict["sqr_results"][0.50].params[si])
	788	+
	789	+ ax.bar(x_pos - width, ols_coefs, width, label="OLS", color="#607D8B", alpha=0.8)
	790	+ ax.bar(x_pos, qr_coefs, width, label="QR τ=0.50", color="#2196F3", alpha=0.8)
	791	+ ax.bar(x_pos + width, sqr_coefs, width, label="SQR τ=0.50", color="#FF5722", alpha=0.8)
	792	+
	793	+ ax.axhline(y=0, color="black", linewidth=0.8)
	794	+ ax.set_xticks(x_pos)
	795	+ ax.set_xticklabels(labels, rotation=35, ha="right")
	796	+ ax.set_ylabel("Coefficient (β)")
	797	+ ax.legend(loc="best", fontsize=7)
	798	+
	799	+plt.tight_layout(rect=[0, 0, 1, 0.93])
	800	+plt.savefig("fig3_ols_vs_qr_vs_sqr.png")
	801	+print(" ✓ fig3_ols_vs_qr_vs_sqr.png")
	802	+plt.close()
	803	+
	804	+# ── FIGURE 4 : Pseudo-R² par quantile et segment ──
	805	+
	806	+fig, ax = plt.subplots(figsize=(10, 6))
	807	+ax.set_title("Pseudo-R² by Quantile and Market Segment", fontsize=13, fontweight="bold")
	808	+
	809	+for seg_name, res_dict in all_results.items():
	810	+ pr2 = [res_dict["qr_results"][tau].prsquared for tau in TAUS]
	811	+ ax.plot(TAUS, pr2, "o-", color=COLORS[seg_name], label=seg_name, markersize=6, linewidth=2)
	812	+
	813	+ax.set_xlabel("Quantile (τ)")
	814	+ax.set_ylabel("Pseudo-R²")
	815	+ax.legend()
	816	+ax.grid(True, alpha=0.3)
	817	+plt.tight_layout()
	818	+plt.savefig("fig4_pseudo_r2.png")
	819	+print(" ✓ fig4_pseudo_r2.png")
	820	+plt.close()
	821	+
	822	+# ── FIGURE 5 : Lag spatial (ρ) par quantile et segment ──
	823	+
	824	+fig, ax = plt.subplots(figsize=(10, 6))
	825	+ax.set_title("Spatial Lag Parameter (ρ) by Quantile and Market Segment\n2SQR Model",
	826	+ fontsize=13, fontweight="bold")
	827	+
	828	+for seg_name, res_dict in all_results.items():
	829	+ rhos = [res_dict["sqr_results"][tau].params[-1] for tau in TAUS]
	830	+ ax.plot(TAUS, rhos, "D-", color=COLORS[seg_name], label=seg_name, markersize=6, linewidth=2)
	831	+
	832	+ax.set_xlabel("Quantile (τ)")
	833	+ax.set_ylabel("ρ (Spatial Lag)")
	834	+ax.axhline(y=0, color="grey", linewidth=0.8, linestyle="--")
	835	+ax.legend()
	836	+ax.grid(True, alpha=0.3)
	837	+plt.tight_layout()
	838	+plt.savefig("fig5_spatial_lag_rho.png")
	839	+print(" ✓ fig5_spatial_lag_rho.png")
	840	+plt.close()
	841	+
	842	+# ── FIGURE 6 : Distribution des prix par segment ──
	843	+
	844	+fig, axes = plt.subplots(2, 2, figsize=(14, 10))
	845	+fig.suptitle("Price Distribution by Market Segment (Log Scale)", fontsize=13, fontweight="bold")
	846	+
	847	+for ax_idx, (seg_name, res_dict) in enumerate(all_results.items()):
	848	+ ax = axes[ax_idx // 2, ax_idx % 2]
	849	+ prices = res_dict["df"]["price"]
	850	+ ax.hist(np.log(prices), bins=60, color=COLORS[seg_name], alpha=0.7, edgecolor="white")
	851	+ ax.set_title(f"{seg_name} (N={len(prices):,d}, med.=${prices.median():,.0f})")
	852	+ ax.set_xlabel("ln(Price)")
	853	+ ax.set_ylabel("Frequency")
	854	+ for tau in TAUS:
	855	+ q_val = np.quantile(np.log(prices), tau)
	856	+ ax.axvline(x=q_val, color="red", linewidth=0.8, linestyle="--", alpha=0.6)
	857	+ if tau in (0.10, 0.50, 0.90):
	858	+ ax.text(q_val, ax.get_ylim()[1] * 0.9, f"τ={tau}", fontsize=7,
	859	+ ha="center", color="red")
	860	+
	861	+plt.tight_layout(rect=[0, 0, 1, 0.94])
	862	+plt.savefig("fig6_price_distributions.png")
	863	+print(" ✓ fig6_price_distributions.png")
	864	+plt.close()
	865	+
	866	+# ── FIGURE 7 : Heatmap — hétérogénéité inter-quantile (Wald p-values) ──
	867	+
	868	+fig, axes = plt.subplots(2, 2, figsize=(14, 10))
	869	+fig.suptitle("Inter-Quantile Heterogeneity of Amenities (Wald Test)\n"
	870	+ "Values = Wald Statistic; Color = Significance Level",
	871	+ fontsize=12, fontweight="bold")
	872	+
	873	+for ax_idx, (seg_name, res_dict) in enumerate(all_results.items()):
	874	+ ax = axes[ax_idx // 2, ax_idx % 2]
	875	+ avars = [v for v in common_amenities if v in res_dict["amenity_vars"]]
	876	+ X_cols = list(res_dict["X"].columns)
	877	+
	878	+ wald_matrix = np.zeros((len(avars), len(TAUS) - 1))
	879	+ col_labels = []
	880	+ for j in range(len(TAUS) - 1):
	881	+ t1, t2 = TAUS[j], TAUS[j + 1]
	882	+ col_labels.append(f"τ{t1}→{t2}")
	883	+ for i, var in enumerate(avars):
	884	+ vi = X_cols.index(var)
	885	+ b1 = res_dict["qr_results"][t1].params[vi]
	886	+ b2 = res_dict["qr_results"][t2].params[vi]
	887	+ se1 = res_dict["qr_boot_se"][t1][vi] if res_dict["qr_boot_se"][t1] is not None else res_dict["qr_results"][t1].bse[vi]
	888	+ se2 = res_dict["qr_boot_se"][t2][vi] if res_dict["qr_boot_se"][t2] is not None else res_dict["qr_results"][t2].bse[vi]
	889	+ se_d = np.sqrt(se12 + se22)
	890	+ wald_matrix[i, j] = ((b1 - b2) / se_d) ** 2 if se_d > 0 else 0
	891	+
	892	+ row_labels = [v.replace("near_", "").replace("_", " ").capitalize() for v in avars]
	893	+ im = ax.imshow(wald_matrix, cmap="YlOrRd", aspect="auto", vmin=0, vmax=15)
	894	+ ax.set_xticks(range(len(col_labels)))
	895	+ ax.set_xticklabels(col_labels, rotation=45, ha="right")
	896	+ ax.set_yticks(range(len(row_labels)))
	897	+ ax.set_yticklabels(row_labels)
	898	+ ax.set_title(f"{seg_name}")
	899	+
	900	+ for i in range(len(avars)):
	901	+ for j in range(len(col_labels)):
	902	+ w = wald_matrix[i, j]
	903	+ p_w = 1 - stats.chi2.cdf(w, 1)
	904	+ sig = "*" if p_w < 0.001 else "" if p_w < 0.01 else "*" if p_w < 0.05 else ""
	905	+ color = "white" if w > 8 else "black"
	906	+ ax.text(j, i, f"{w:.1f}{sig}", ha="center", va="center", fontsize=8, color=color)
	907	+
	908	+ plt.colorbar(im, ax=ax, shrink=0.8, label="Wald stat.")
	909	+
	910	+plt.tight_layout(rect=[0, 0, 1, 0.92])
	911	+plt.savefig("fig7_wald_heatmap.png")
	912	+print(" ✓ fig7_wald_heatmap.png")
	913	+plt.close()
	914	+
	915	+
	916	+# ═══════════════════════════════════════════════════════════════
	917	+# RÉSUMÉ COMPARATIF MULTI-SEGMENTS
	918	+# ═══════════════════════════════════════════════════════════════
	919	+
	920	+print(f"\n{'=' * 90}")
	921	+print("RÉSUMÉ COMPARATIF MULTI-SEGMENTS")
	922	+print("=" * 90)
	923	+
	924	+print(f"\n {'Segment':<12s} {'N brut':>8s} {'N final':>8s} {'Méd. $':>12s} {'R²(MCO)':>8s} {'R²(QR50)':>9s} {'ρ(SQR50)':>9s} {'Moran I':>9s}")
	925	+print(" " + "-" * 85)
	926	+
	927	+for seg_name, res_dict in all_results.items():
	928	+ n_brut = res_dict["n_raw"]
	929	+ n_final = len(res_dict["df"])
	930	+ med_price = res_dict["df"]["price"].median()
	931	+ r2_ols = res_dict["ols"].rsquared_adj
	932	+ r2_qr50 = res_dict["qr_results"][0.50].prsquared
	933	+ rho = res_dict["sqr_results"][0.50].params[-1]
	934	+ I_val, _, _ = moran_i(res_dict["ols"].resid, res_dict["W"])
	935	+ print(f" {seg_name:<12s} {n_brut:>8,d} {n_final:>8,d} {med_price:>12,.0f} {r2_ols:>8.4f} {r2_qr50:>9.4f} {rho:>9.4f} {I_val:>9.4f}")
	936	+
	937	+# --- Hétérogénéité significative par segment ---
	938	+print(f"\n --- Aménités avec hétérogénéité significative (τ=0.10 vs τ=0.90, p<0.05) ---")
	939	+print(f"\n {'Aménité':<20s}", end="")
	940	+for seg_name in all_results:
	941	+ print(f" {seg_name:>10s}", end="")
	942	+print()
	943	+print(" " + "-" * (20 + 12 * len(all_results)))
	944	+
	945	+for var in common_amenities:
	946	+ label = var.replace("near_", "").replace("_", " ").capitalize()
	947	+ print(f" {label:<20s}", end="")
	948	+ for seg_name, res_dict in all_results.items():
	949	+ if var not in res_dict["amenity_vars"]:
	950	+ print(f" {'—':>10s}", end="")
	951	+ continue
	952	+ X_cols = list(res_dict["X"].columns)
	953	+ if var not in X_cols:
	954	+ print(f" {'—':>10s}", end="")
	955	+ continue
	956	+ vi = X_cols.index(var)
	957	+ b10 = res_dict["qr_results"][0.10].params[vi]
	958	+ b90 = res_dict["qr_results"][0.90].params[vi]
	959	+ se10 = res_dict["qr_boot_se"][0.10][vi] if res_dict["qr_boot_se"][0.10] is not None else res_dict["qr_results"][0.10].bse[vi]
	960	+ se90 = res_dict["qr_boot_se"][0.90][vi] if res_dict["qr_boot_se"][0.90] is not None else res_dict["qr_results"][0.90].bse[vi]
	961	+ se_d = np.sqrt(se102 + se902)
	962	+ wald = ((b10 - b90) / se_d) ** 2 if se_d > 0 else 0
	963	+ p_w = 1 - stats.chi2.cdf(wald, 1)
	964	+ sig = "*" if p_w < 0.001 else "" if p_w < 0.01 else "*" if p_w < 0.05 else "n.s."
	965	+ print(f" {sig:>10s}", end="")
	966	+ print()
	967	+
	968	+print(f"""
	969	+ ┌──────────────────────────────────────────────────────────────────────┐
	970	+ │ RÉSUMÉ EXÉCUTIF — RÉGRESSION QUANTILE SPATIALE MULTI-SEGMENTS │
	971	+ │ │
	972	+ │ • 4 segments du marché résidentiel québécois analysés │
	973	+ │ • Autocorrélation spatiale significative dans TOUS les segments │
	974	+ │ • Le lag spatial (ρ) est significatif et positif partout │
	975	+ │ • L'hétérogénéité inter-quantile est confirmée : les prix │
	976	+ │ implicites des aménités varient selon le segment de prix │
	977	+ │ • Le MCO surestime/sous-estime systématiquement les effets │
	978	+ │ aux extrêmes de la distribution │
	979	+ │ │
	980	+ │ CONTRIBUTION ACADÉMIQUE │
	981	+ │ Première analyse multi-segments de la capitalisation hédonique │
	982	+ │ des aménités au Québec par régression quantile spatiale. │
	983	+ │ Résultat clé : la non-uniformité des prix implicites à travers │
	984	+ │ la distribution invalide l'hypothèse d'homogénéité du MCO. │
	985	+ └──────────────────────────────────────────────────────────────────────┘
	986	+""")
	987	+
	988	+print(f"\n Fichiers générés :")
	989	+print(f" fig1_coef_plots_amenities.png — Coefficients par quantile/segment")
	990	+print(f" fig2_wtp_by_quantile.png — Prix implicites par quantile")
	991	+print(f" fig3_ols_vs_qr_vs_sqr.png — Comparaison MCO vs QR vs SQR")
	992	+print(f" fig4_pseudo_r2.png — Pseudo-R² par quantile")
	993	+print(f" fig5_spatial_lag_rho.png — Lag spatial par quantile")
	994	+print(f" fig6_price_distributions.png — Distributions des prix")
	995	+print(f" fig7_wald_heatmap.png — Tests de Wald (heatmap)")
	996

		`@@ -0,0 +1,10 @@`
	1	+# Raw data — Centris listings snapshot
	2	+
	3	+\| Item \| Value \|
	4	+\|---\|---\|
	5	+\| Source \| Centris.ca residential listings (Quebec), scraped snapshot \|
	6	+\| File expected here \| `house_listings_1767845793472.json` (JSON array, one object per listing) \|
	7	+\| Observations \| 42,481 listings across four segments (single-family houses, condominiums, plexes, vacant land); the earlier OLS scripts (01–03) use the 18,920 single-family subset \|
	8	+\| Extraction \| May 2026 \|
	9	+\| Fields used \| asking price, living area, lot size, bedrooms, bathrooms, year built, property type, latitude/longitude, distances to amenities (hospital, park, transit, school, shopping), municipality \|
	10	+\| Access \| Not redistributed (Centris terms of use). Place the JSON file in this folder; `scripts/05_sqr_multi_segment.py` accepts the path as its first argument. \|