WP10 — The Assessment Gap in Quebec: full research project · 2fb3ff1 · wp10_uqo

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++# Raw data snapshot — 100 MB parquet, not redistributed
++data/raw/*.parquet
++data/processed/*.parquet
++
++# Python
++__pycache__/
++*.pyc
++
++# LaTeX build artifacts (paper/main.pdf IS committed)
++paper/*.aux
++paper/*.log
++paper/*.out
++paper/*.fls
++paper/*.fdb_latexmk
++paper/*.bbl
++paper/*.blg
++paper/*.synctex.gz
++
++# macOS
++.DS_Store

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++<!-- Author: Simon-Pierre Boucher — contact@spboucher.ai -->
++# ⚖️ The Assessment Gap in Quebec
++
++**UQO Working Paper No. 10** — *Vertical and Horizontal Inequity in Municipal
++Property Valuation: Evidence from 522,769 Sales Matched to the Assessment Roll*
++
++[![Paper](https://img.shields.io/badge/paper-PDF%20(29%20p.)-b31b1b?logo=latex&logoColor=white)](paper/main.pdf)
++[![Python](https://img.shields.io/badge/python-3.11%2B-3776AB?logo=python&logoColor=white)](requirements.txt)
++[![Data](https://img.shields.io/badge/data-745%2C119%20matched%20sales%20·%20parquet-4051b5)](data/raw/README.md)
++[![Estimation](https://img.shields.io/badge/estimation-statsmodels%20·%20linearmodels%20(AbsorbingLS%20%2B%20IV2SLS)-4051b5)](src/wp10/models.py)
++[![Reproducible](https://img.shields.io/badge/reproducible-end--to--end%20pipeline-2e7d32)](scripts/)
++[![Institution](https://img.shields.io/badge/UQO-D%C3%A9pt.%20des%20sciences%20administratives-16365c)](https://uqo.ca)
++[![Author](https://img.shields.io/badge/author-Simon--Pierre%20Boucher-16365c)](mailto:contact@spboucher.ai)
++[![Contact](https://img.shields.io/badge/contact-contact%40spboucher.ai-a02020?logo=maildotru&logoColor=white)](mailto:contact@spboucher.ai)
++
++> **TL;DR** — Matching 522,769 Quebec residential sales (2021–2026) to the triennial
++> assessment rolls that tax them, this paper delivers the **first province-wide audit
++> of property-assessment equity in Canada**. Within the same municipality × roll ×
++> year, the elasticity of the assessment ratio w.r.t. price is **−0.34** (Cheng FE)
++> and still **−0.08** under Clapp's measurement-error-robust IV. **99% of
++> municipalities** fail the IAAO uniformity standard, **95%** fail its vertical-equity
++> standard — Montréal is the lone progressive large market. The median dwelling in the
++> bottom local price decile pays **~65% more property tax** than uniform assessment
++> would imply; the top decile pays ~5% less.
++
++---
++
++## 📖 Table of contents
++
++1. [What this paper does](#-what-this-paper-does)
++2. [Headline results](#-headline-results)
++3. [Repository layout](#-repository-layout)
++4. [Quick start](#-quick-start)
++5. [The pipeline, step by step](#-the-pipeline-step-by-step)
++6. [Data](#-data)
++7. [Methodology](#-methodology)
++8. [Figures & tables inventory](#-figures--tables-inventory)
++9. [Limitations](#-limitations)
++10. [Citation](#-citation)
++11. [Author & contact](#-author--contact)
++
++---
++
++## 🎯 What this paper does
++
++Quebec taxes every dwelling in proportion to its **assessed value**, redrawn on a
++triennial roll whose values must, by statute (art. 46 LFM), reflect market conditions
++at a single reference date. If assessments are *regressive* — cheap homes overvalued
++relative to expensive ones — the effective tax rate falls with wealth, silently.
++This project measures that gap at province scale:
++
++```
++ln AV_i = α_c(i) + β · ln SP_i + ε_i      (c = municipality × roll × sale-year)
++```
++
++- **γ = β − 1** is the elasticity of the assessment ratio w.r.t. price
++  (γ < 0 ⇒ regressive), identified **within** 2,884 market-timing cells;
++- the **Clapp (1990) rank IV** purges the attenuation bias that pushes naïve
++  ratio studies toward spurious regressivity;
++- the full **IAAO ratio-study battery** (median ratio, COD, PRD, PRB) with
++  bootstrap CIs is computed for every municipality with ≥ 100 usable sales;
++- quantile profiles, subgroup elasticities, horizontal-dispersion regressions and
++  an exact **intra-municipal tax-shift** calculation complete the audit.
++
++## 🏆 Headline results
++
++| Object | Estimate |
++|---|---|
++| Cheng elasticity, pooled OLS | β = 0.812 (0.029) |
++| Cheng elasticity, cell FE | **β = 0.656 (0.029)** → γ = −0.344 |
++| Clapp rank-IV (error-robust) | **β = 0.917 (0.021)** → γ = −0.083 |
++| Quantile profile β(τ), τ = .10 → .90 | 0.87 → **0.49** (failure concentrated at the top) |
++| Province-wide PRB (2021 / 2024) | −0.104 / −0.002 (staleness mechanism) |
++| Municipalities with PRB < 0 | **99%** (95% below IAAO band) |
++| Median municipality COD | 26 (IAAO ceiling: 15) |
++| Excess tax, bottom ↔ top local decile | **+65% ↔ −5%** (median) |
++| The exception | Montréal: PRB **+0.07**, progressive in all 6 years |
++
++Sub-group anatomy: condos γ = −0.12 vs single-family −0.45 / plex −0.49;
++land share > 0.4 → −0.50; age > 60 y → −0.43; large markets (>10k sales) −0.27.
++
++## 🗂 Repository layout
++
++```
++wp10_uqo/
++├── data/
++│   ├── raw/                  # matched sales–roll snapshot (see data/raw/README.md)
++│   └── processed/            # analysis.parquet (built by step 01)
++├── src/wp10/                 # reusable package
++│   ├── config.py             # paths, thresholds, constants
++│   ├── sample.py             # sample construction & variable definitions
++│   ├── iaao.py               # COD / PRD / PRB + bootstrap CIs
++│   ├── models.py             # Cheng FE, Clapp IV, quantiles, horizontal, tax shift
++│   └── plotstyle.py          # journal-calibre matplotlib style (validated palette)
++├── scripts/                  # the pipeline, in order
++│   ├── 01_build_sample.py
++│   ├── 02_iaao_stats.py
++│   ├── 03_estimate_regressions.py
++│   ├── 04_make_figures.py
++│   └── 05_make_tables.py
++├── results/
++│   ├── reproduced/           # every CSV the scripts emit
++│   └── tables/               # LaTeX tables input by the paper
++├── figures/                  # 10 publication figures (PNG, 300 dpi)
++└── paper/                    # main.tex + sections/ + references.bib → main.pdf
++```
++
++## 🚀 Quick start
++
++```bash
++cd wp10_uqo
++python3 -m pip install -r requirements.txt
++
++# place the parquet snapshot in data/raw/ (see data/raw/README.md), then:
++python3 scripts/01_build_sample.py         # 745,119 → 522,769 sales
++python3 scripts/02_iaao_stats.py           # IAAO diagnostics + bootstrap
++python3 scripts/03_estimate_regressions.py # FE / IV / quantile / heterogeneity / tax shift
++python3 scripts/04_make_figures.py         # 10 figures
++python3 scripts/05_make_tables.py          # 10 LaTeX tables
++
++cd paper && make                           # compile main.pdf (latexmk)
++```
++
++Total runtime ≈ 5 minutes on an Apple-silicon laptop; peak RAM ≈ 6 GB.
++
++## 🗃 Data
++
++745,119 residential-market transactions (Jan 2021 – Jul 2026) matched at the
++**parcel level** to the assessment roll in force at the sale date (median match
++distance 0.6 m; every retained sale reproduces the roll value exactly). From the
++roll: total/land/building assessed values, lot & floor areas, year built, unit
++count, CUBF use code, roll vintage and the statutory market-condition date.
++Sample cascade (details in the paper, §4): residential CUBF → match quality →
++valid AV/SP → 1/99% ratio trim within roll vintage → cells ≥ 20 sales
++⇒ **522,769 sales, 625 municipalities, 2,884 cells**.
++
++## 🔬 Methodology
++
++1. **IAAO diagnostics** (Standard on Ratio Studies, 2013) within
++   municipality × sale-year blocks, aggregated by median — COD, PRD, PRB with
++   percentile-bootstrap CIs (`src/wp10/iaao.py`).
++2. **Cheng (1974) log-log regression** with 2,884 absorbed fixed effects
++   (linearmodels `AbsorbingLS`), SEs clustered on 625 municipalities.
++3. **Clapp (1990) rank IV** — Z ∈ {−1,0,+1} from within-cell rank agreement of
++   ln AV and ln SP, 2SLS on demeaned data (`IV2SLS`) — the conservative bound.
++4. **Quantile regressions** on within-cell demeaned data (τ = .10….90).
++5. **Heterogeneity**: separate FE estimates by class, age, land share, roll lag,
++   municipality size, sale year.
++6. **Horizontal inequity**: |ln r − cell median| regressed on characteristics.
++7. **Tax shift**: r / cell-median − 1 by within-cell price decile — exact %
++   over/under-payment under Quebec's exemption-free ad valorem rule.
++
++## 🖼 Figures & tables inventory
++
++| Figure | Content |
++|---|---|
++| `fig_ratio_dist` | ratio distribution + by roll lag (Panels A/B) |
++| `fig_time` | median ratio by month × roll vintage (staleness drift) |
++| `fig_binscatter` | within-cell ln ratio vs ln price, 20 bins — the core fact |
++| `fig_quantile` | β(τ) profile with FE/IV benchmarks |
++| `fig_heterogeneity` | forest plot of subgroup γ |
++| `fig_prb_muni`, `fig_cod` | municipal PRB / COD distributions vs IAAO bands |
++| `fig_map` | province map of municipal PRB (Montréal exception) |
++| `fig_taxshift` | excess tax burden by local price decile |
++| `fig_robustness` | γ_FE vs γ_IV across 10 sample variants |
++
++Tables (`results/tables/*.tex`): summary stats, IAAO province-wide, ten largest
++markets, vertical regressions, quantiles, heterogeneity, horizontal, tax shift,
++robustness, Paglin–Fogarty.
++
++## ⚠️ Limitations
++
++- Sales < $50k absent from the source registry; residual non-arm's-length
++  transfers may survive screens (bottom-decile shift = upper bound).
++- 2021–2026 is an unusually turbulent market — levels statistics are
++  period-specific (the within-market elasticity is not).
++- No appeals, renovations, or owner demographics observed.
++- Montréal's progressivity may partly reflect borough-level composition.
++
++## 📝 Citation
++
++```bibtex
++@techreport{boucher2026assessmentgap,
++  author      = {Boucher, Simon-Pierre},
++  title       = {The Assessment Gap in Quebec: Vertical and Horizontal Inequity
++                 in Municipal Property Valuation},
++  institution = {Universit\'e du Qu\'ebec en Outaouais,
++                 D\'epartement des sciences administratives},
++  type        = {Working Paper},
++  number      = {10},
++  year        = {2026},
++  month       = {August}
++}
++```
++
++## 👤 Author & contact
++
++**Simon-Pierre Boucher** — Département des sciences administratives,
++Université du Québec en Outaouais (UQO), Gatineau, QC.
++📧 [contact@spboucher.ai](mailto:contact@spboucher.ai)

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++<!-- Author: Simon-Pierre Boucher — contact@spboucher.ai -->
++# Raw data
++
++`transactions_700k_avec_registre_foncier.parquet` — 745,119 Quebec
++residential-market transactions (January 2021 – July 2026) matched at the
++parcel level to the municipal assessment roll (rôle d'évaluation foncière)
++in force at the sale date.
++
++| Group | Key columns |
++|---|---|
++| Transaction | `id`, `date`, `amount`, `street`, `zipCode`, `city`, `lat`, `lng`, `propertyType`, `ownerType`, `tx_year` |
++| Roll identity | `role_id_provinc`, `role_matricule`, `role_code_mun`, `role_municipalite`, `role_anrole` (roll vintage), `role_date_cond_marche` (statutory market-condition reference date) |
++| Roll values | `role_valeur_immeuble` (total AV), `role_valeur_terrain` (land), `role_valeur_batiment` (building), `role_valeur_role_anterieur`, `previousValue`, `totalArValue` |
++| Roll structure | `role_cubf` (+ `role_cubf_libelle`), `role_superficie_terrain_m2`, `role_aire_etages_m2`, `role_annee_construction`, `role_nb_logements`, `role_nb_etages`, `role_lien_physique`, `role_genre_construction`, `role_unite_voisinage` |
++| Match quality | `match_dist_m` (median 0.6 m), `match_score` (max 220), `match_valeur_exacte` |
++
++The file is **not redistributed** with the repository (100 MB; see
++`.gitignore`). Place it in this directory before running the pipeline.

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++# Author: Simon-Pierre Boucher — contact@spboucher.ai
++$pdf_mode = 1;
++$pdflatex = 'pdflatex -interaction=nonstopmode -halt-on-error -synctex=1 %O %S';
++$bibtex_use = 2;
++$clean_ext = 'synctex.gz run.xml bbl bcf fdb_latexmk fls log aux out toc lof lot blg';
++@default_files = ('main.tex');

added paper/Makefile +9 −0

++# Author: Simon-Pierre Boucher — contact@spboucher.ai
++all:
++	latexmk -pdf main.tex
++
++clean:
++	latexmk -c
++
++distclean:
++	latexmk -C

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++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++% UQO Working Paper No. 10
++% The Assessment Gap in Quebec: Vertical and Horizontal Inequity in
++% Municipal Property Valuation
++%
++% Build:  latexmk -pdf main.tex   (or `make` in this directory)
++% Figures are read from ../figures/, tables from ../results/tables/.
++% ============================================================================
++\documentclass[12pt,letterpaper]{article}
++
++% ---------------------------------------------------------------- encoding
++\usepackage[utf8]{inputenc}
++\usepackage[T1]{fontenc}
++\usepackage[english]{babel}
++
++% ---------------------------------------------------------------- layout
++\usepackage[letterpaper,margin=1in]{geometry}
++\usepackage{setspace}
++\onehalfspacing
++
++% ---------------------------------------------------------------- typography
++\usepackage{mathptmx}
++\usepackage{microtype}
++
++% ---------------------------------------------------------------- math
++\usepackage{amsmath,amssymb,amsthm}
++
++% ---------------------------------------------------------------- tables
++\usepackage{booktabs}
++\usepackage{threeparttable}
++\usepackage{makecell}
++
++% ---------------------------------------------------------------- figures
++\usepackage{graphicx}
++\usepackage{subcaption}
++\graphicspath{{../figures/}{./}}
++
++% ---------------------------------------------------------------- captions
++\usepackage[font=small,labelfont=bf,labelsep=period,justification=justified,singlelinecheck=false]{caption}
++
++% ---------------------------------------------------------------- colours & links
++\usepackage[dvipsnames]{xcolor}
++\usepackage[colorlinks=true,linkcolor=NavyBlue,citecolor=NavyBlue,urlcolor=NavyBlue,breaklinks=true]{hyperref}
++
++% ---------------------------------------------------------------- bibliography
++\usepackage[authoryear,round,semicolon]{natbib}
++\bibliographystyle{aer}
++
++% ---------------------------------------------------------------- headings & lists
++\usepackage{titlesec}
++\titleformat{\section}{\large\bfseries}{\thesection.}{0.5em}{}
++\titleformat{\subsection}{\normalsize\bfseries}{\thesubsection.}{0.5em}{}
++\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection.}{0.5em}{}
++\usepackage{fancyhdr}
++\pagestyle{fancy}\fancyhf{}
++\renewcommand{\headrulewidth}{0pt}
++\fancyfoot[C]{\thepage}
++\usepackage{enumitem}
++\setlist{nosep,leftmargin=*}
++
++% ============================================================================
++% METADATA
++% ============================================================================
++\newcommand{\WPnumber}{10}
++\newcommand{\WPtitle}{The Assessment Gap in Quebec}
++\newcommand{\WPsubtitle}{Vertical and Horizontal Inequity in Municipal Property
++  Valuation: Evidence from 522{,}769 Sales Matched to the Assessment Roll}
++\newcommand{\WPdate}{August 2026}
++\newcommand{\WPversion}{1.0}
++\newcommand{\WPkeywords}{Property tax, Assessment, Vertical equity,
++  Regressivity, Ratio study, Quebec}
++\newcommand{\WPjel}{H71, H22, R51, R31}
++
++\newcommand{\WPauthor}{Simon-Pierre Boucher}
++\newcommand{\WPaffiliation}{D\'epartement des sciences administratives\\
++  Universit\'e du Qu\'ebec en Outaouais}
++\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}
++\newcommand{\WPaddress}{Gatineau -- Pavillon Alexandre-Tach\'e\\
++  283, boulevard Alexandre-Tach\'e\\ Gatineau, Qu\'ebec, Canada J9A 1L8}
++
++\newcommand{\WPabstract}{%
++Municipal property taxes are levied on assessed values, so any systematic
++relationship between assessment accuracy and market value silently
++redistributes the tax burden. Matching 522{,}769 residential transactions
++from 2021--2026 to the triennial assessment rolls of 625 Quebec
++municipalities at the parcel level, we provide the first province-wide
++evaluation of vertical and horizontal equity in Canadian property assessment.
++Assessment ratios decline steeply with price: within
++municipality~$\times$~roll~$\times$~sale-year markets, the elasticity of the
++assessment ratio with respect to the sale price is $-0.34$, and a rank-based
++instrumental-variable estimator in the spirit of \mbox{Clapp (1990)}---which
++purges the attenuation bias that pushes least-squares tests toward spurious
++regressivity---still yields $-0.08$ ($t\approx4$). IAAO ratio-study
++diagnostics place 95\% of municipalities below the acceptable price-related
++bias band and 99\% above the COD uniformity ceiling; Montr\'eal is the only
++large market whose assessments are progressive. Regressivity is strongest for
++single-family homes, older buildings, properties with a high assessed land
++share, and smaller municipalities, and it doubles in years when market prices
++decouple from the rolls' reference dates, pointing to infrequent revaluation
++as a first-order mechanism. The implied intra-municipal tax shift is large:
++the median dwelling in the bottom within-market price decile pays roughly
++65\% more property tax than it would under uniform assessment, while the top
++decile pays about 5\% less.}
++
++\begin{document}
++
++\input{sections/titlepage}
++
++\setcounter{page}{1}
++\input{sections/introduction}
++\input{sections/literature}
++\input{sections/institutional}
++\input{sections/data}
++\input{sections/methodology}
++\input{sections/results}
++\input{sections/robustness}
++\input{sections/discussion}
++\input{sections/conclusion}
++
++\newpage
++\bibliography{references}
++
++\end{document}

added paper/references.bib +402 −0

++@comment{ Author: Simon-Pierre Boucher (contact at spboucher.ai) -- Bibliography for UQO Working Paper No 10 }
++
++@article{paglin1972equity,
++  author  = {Paglin, Morton and Fogarty, Michael},
++  title   = {Equity and the Property Tax: A New Conceptual Focus},
++  journal = {National Tax Journal},
++  year    = {1972},
++  volume  = {25},
++  number  = {4},
++  pages   = {557--565}
++}
++
++@article{cheng1974property,
++  author  = {Cheng, Pao Lun},
++  title   = {Property Taxation, Assessment Performance, and Its Measurement},
++  journal = {Public Finance},
++  year    = {1974},
++  volume  = {29},
++  number  = {3},
++  pages   = {268--284}
++}
++
++@article{kochin1982vertical,
++  author  = {Kochin, Levis A. and Parks, Richard W.},
++  title   = {Vertical Equity in Real Estate Assessment: A Fair Appraisal},
++  journal = {Economic Inquiry},
++  year    = {1982},
++  volume  = {20},
++  number  = {4},
++  pages   = {511--532}
++}
++
++@article{bell1984administrative,
++  author  = {Bell, Earl J.},
++  title   = {Administrative Inequity and Property Assessment: The Case for the Traditional Approach},
++  journal = {Property Tax Journal},
++  year    = {1984},
++  volume  = {3},
++  number  = {2},
++  pages   = {123--131}
++}
++
++@article{kennedy1984unfair,
++  author  = {Kennedy, Peter},
++  title   = {On an Unfair Appraisal of Vertical Equity in Real Estate Assessment},
++  journal = {Economic Inquiry},
++  year    = {1984},
++  volume  = {22},
++  number  = {2},
++  pages   = {287--290}
++}
++
++@article{sunderman1990testing,
++  author  = {Sunderman, Mark A. and Birch, John W. and Cannaday, Roger E. and Hamilton, Thomas W.},
++  title   = {Testing for Vertical Inequity in Property Tax Systems},
++  journal = {Journal of Real Estate Research},
++  year    = {1990},
++  volume  = {5},
++  number  = {3},
++  pages   = {319--334}
++}
++
++@article{clapp1990new,
++  author  = {Clapp, John M.},
++  title   = {A New Test for Equitable Real Estate Tax Assessment},
++  journal = {Journal of Real Estate Finance and Economics},
++  year    = {1990},
++  volume  = {3},
++  number  = {3},
++  pages   = {233--249}
++}
++
++@article{sirmans1995vertical,
++  author  = {Sirmans, G. Stacy and Diskin, Barry A. and Friday, H. Swint},
++  title   = {Vertical Inequity in the Taxation of Real Property},
++  journal = {National Tax Journal},
++  year    = {1995},
++  volume  = {48},
++  number  = {1},
++  pages   = {71--84}
++}
++
++@article{allen2002measuring,
++  author  = {Allen, Marcus T. and Dare, William H.},
++  title   = {Identifying Determinants of Horizontal Property Tax Inequity: Evidence from Florida},
++  journal = {Journal of Real Estate Research},
++  year    = {2002},
++  volume  = {24},
++  number  = {2},
++  pages   = {153--164}
++}
++
++@article{sirmans2008vertical,
++  author  = {Sirmans, G. Stacy and Gatzlaff, Dean H. and Macpherson, David A.},
++  title   = {Horizontal and Vertical Inequity in Real Property Taxation},
++  journal = {Journal of Real Estate Literature},
++  year    = {2008},
++  volume  = {16},
++  number  = {2},
++  pages   = {167--180}
++}
++
++@article{mcmillen2008thin,
++  author  = {McMillen, Daniel P. and Weber, Rachel N.},
++  title   = {Thin Markets and Property Tax Inequities: A Multinomial Logit Approach},
++  journal = {National Tax Journal},
++  year    = {2008},
++  volume  = {61},
++  number  = {4},
++  pages   = {653--671}
++}
++
++@article{mcmillen2011assessment,
++  author  = {McMillen, Daniel P.},
++  title   = {Assessment Regressivity: A Tale of Two {Illinois} Counties},
++  journal = {Land Lines},
++  year    = {2011},
++  volume  = {23},
++  number  = {1},
++  pages   = {9--15}
++}
++
++@article{hodge2017assessment,
++  author  = {Hodge, Timothy R. and McMillen, Daniel P. and Sands, Gary and Skidmore, Mark},
++  title   = {Assessment Inequity in a Declining Housing Market: The Case of {Detroit}},
++  journal = {Real Estate Economics},
++  year    = {2017},
++  volume  = {45},
++  number  = {2},
++  pages   = {237--258}
++}
++
++@article{mcmillen2020assessment,
++  author  = {McMillen, Daniel and Singh, Ruchi},
++  title   = {Assessment Regressivity and Property Taxation},
++  journal = {Journal of Real Estate Finance and Economics},
++  year    = {2020},
++  volume  = {60},
++  number  = {1},
++  pages   = {155--169}
++}
++
++@article{avenancio2022assessment,
++  author  = {Avenancio-Le{\'o}n, Carlos F. and Howard, Troup},
++  title   = {The Assessment Gap: Racial Inequalities in Property Taxation},
++  journal = {Quarterly Journal of Economics},
++  year    = {2022},
++  volume  = {137},
++  number  = {3},
++  pages   = {1383--1434}
++}
++
++@techreport{berry2021reassessing,
++  author      = {Berry, Christopher R.},
++  title       = {Reassessing the Property Tax},
++  institution = {University of Chicago, Harris School of Public Policy},
++  type        = {Working Paper},
++  year        = {2021}
++}
++
++@techreport{amornsiripanitch2022residential,
++  author      = {Amornsiripanitch, Natee},
++  title       = {Why Are Residential Property Tax Rates Regressive?},
++  institution = {Federal Reserve Bank of Philadelphia},
++  type        = {Working Paper},
++  number      = {22-02},
++  year        = {2022}
++}
++
++@article{mcmillen2022allocative,
++  author  = {McMillen, Daniel and Singh, Ruchi},
++  title   = {Measures of Vertical Inequality in Assessments},
++  journal = {Journal of Real Estate Finance and Economics},
++  year    = {2022},
++  volume  = {online first},
++  pages   = {1--29}
++}
++
++@book{iaao2013standard,
++  author    = {{International Association of Assessing Officers}},
++  title     = {Standard on Ratio Studies},
++  publisher = {IAAO},
++  address   = {Kansas City, MO},
++  year      = {2013}
++}
++
++@book{eckert1990property,
++  author    = {Eckert, Joseph K. and Gloudemans, Robert J. and Almy, Richard R.},
++  title     = {Property Appraisal and Assessment Administration},
++  publisher = {International Association of Assessing Officers},
++  address   = {Chicago},
++  year      = {1990}
++}
++
++@book{gloudemans2011fundamentals,
++  author    = {Gloudemans, Robert and Almy, Richard},
++  title     = {Fundamentals of Mass Appraisal},
++  publisher = {International Association of Assessing Officers},
++  address   = {Kansas City, MO},
++  year      = {2011}
++}
++
++@article{oates1969effects,
++  author  = {Oates, Wallace E.},
++  title   = {The Effects of Property Taxes and Local Public Spending on Property Values: An Empirical Study of Tax Capitalization and the {Tiebout} Hypothesis},
++  journal = {Journal of Political Economy},
++  year    = {1969},
++  volume  = {77},
++  number  = {6},
++  pages   = {957--971}
++}
++
++@article{oates2016local,
++  author  = {Oates, Wallace E. and Fischel, William A.},
++  title   = {Are Local Property Taxes Regressive, Progressive, or What?},
++  journal = {National Tax Journal},
++  year    = {2016},
++  volume  = {69},
++  number  = {2},
++  pages   = {415--434}
++}
++
++@article{rosen1974hedonic,
++  author  = {Rosen, Sherwin},
++  title   = {Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition},
++  journal = {Journal of Political Economy},
++  year    = {1974},
++  volume  = {82},
++  number  = {1},
++  pages   = {34--55}
++}
++
++@incollection{malpezzi2003hedonic,
++  author    = {Malpezzi, Stephen},
++  title     = {Hedonic Pricing Models: A Selective and Applied Review},
++  booktitle = {Housing Economics and Public Policy},
++  editor    = {O'Sullivan, Tony and Gibb, Kenneth},
++  publisher = {Blackwell},
++  address   = {Oxford},
++  year      = {2003},
++  pages     = {67--89}
++}
++
++@article{carbonnier2024property,
++  author  = {Carbonnier, Cl{\'e}ment},
++  title   = {Property Tax Regressivity, the Case of {Qu{\'e}bec}},
++  journal = {Public Finance Review},
++  year    = {2024},
++  volume  = {52},
++  number  = {2},
++  pages   = {155--181}
++}
++
++@techreport{slack2014political,
++  author      = {Slack, Enid and Bird, Richard M.},
++  title       = {The Political Economy of Property Tax Reform},
++  institution = {OECD},
++  type        = {OECD Working Papers on Fiscal Federalism},
++  number      = {18},
++  year        = {2014}
++}
++
++@book{youngman2016good,
++  author    = {Youngman, Joan},
++  title     = {A Good Tax: Legal and Policy Issues for the Property Tax in the {United States}},
++  publisher = {Lincoln Institute of Land Policy},
++  address   = {Cambridge, MA},
++  year      = {2016}
++}
++
++@techreport{found2017property,
++  author      = {Found, Adam and Tomlinson, Peter},
++  title       = {Business Tax Burdens in {Canada's} Major Cities: The 2017 Report Card},
++  institution = {C.D. Howe Institute},
++  type        = {E-Brief},
++  year        = {2017}
++}
++
++@techreport{mpac2022vertical,
++  author      = {{Municipal Property Assessment Corporation}},
++  title       = {Vertical Equity Review of Residential Assessed Values},
++  institution = {MPAC},
++  address     = {Toronto},
++  type        = {Technical Report},
++  year        = {2022}
++}
++
++@article{goolsby1997assessment,
++  author  = {Goolsby, William C.},
++  title   = {Assessment Error in the Valuation of Owner-Occupied Housing},
++  journal = {Journal of Real Estate Research},
++  year    = {1997},
++  volume  = {13},
++  number  = {1},
++  pages   = {33--45}
++}
++
++@article{quintos2020prb,
++  author  = {Quintos, Carmela},
++  title   = {A Gradient Boosting Approach to Understanding Vertical and Horizontal Inequity in Property Assessments},
++  journal = {Journal of Property Tax Assessment \& Administration},
++  year    = {2020},
++  volume  = {17},
++  number  = {2},
++  pages   = {1--28}
++}
++
++@article{ross2012assessor,
++  author  = {Ross, Justin M.},
++  title   = {Interjurisdictional Determinants of Property Assessment Regressivity},
++  journal = {Land Economics},
++  year    = {2012},
++  volume  = {88},
++  number  = {1},
++  pages   = {28--42}
++}
++
++@article{plummer2014evidence,
++  author  = {Plummer, Elizabeth},
++  title   = {The Effects of Property Tax Protests on the Assessment Uniformity of Residential Properties},
++  journal = {Real Estate Economics},
++  year    = {2014},
++  volume  = {42},
++  number  = {4},
++  pages   = {900--937}
++}
++
++@article{clapp1990methods,
++  author  = {Clapp, John M.},
++  title   = {A Methodology for Constructing Vacant Land Price Indices},
++  journal = {Real Estate Economics},
++  year    = {1990},
++  volume  = {18},
++  number  = {3},
++  pages   = {274--293}
++}
++
++@misc{lfm2026,
++  author       = {{Gouvernement du Qu\'ebec}},
++  title        = {Loi sur la fiscalit\'e municipale, {RLRQ}, c.\ {F-2.1}},
++  howpublished = {L\'egisQu\'ebec},
++  year         = {2026},
++  note         = {Articles 42--46 (triennial rolls and market-condition reference date)}
++}
++
++@misc{mamh2024manuel,
++  author       = {{Minist\`ere des Affaires municipales et de l'Habitation}},
++  title        = {Manuel d'\'evaluation fonci\`ere du {Qu\'ebec}},
++  howpublished = {Gouvernement du Qu\'ebec},
++  address      = {Qu\'ebec},
++  year         = {2024}
++}
++
++@article{desrosiers2000hedonic,
++  author  = {Des Rosiers, Fran{\c c}ois and Th{\'e}riault, Marius and Villeneuve, Paul-Y.},
++  title   = {Sorting Out Access and Neighbourhood Factors in Hedonic Price Modelling},
++  journal = {Journal of Property Investment \& Finance},
++  year    = {2000},
++  volume  = {18},
++  number  = {3},
++  pages   = {291--315}
++}
++
++@article{bostic2007land,
++  author  = {Bostic, Raphael W. and Longhofer, Stanley D. and Redfearn, Christian L.},
++  title   = {Land Leverage: Decomposing Home Price Dynamics},
++  journal = {Real Estate Economics},
++  year    = {2007},
++  volume  = {35},
++  number  = {2},
++  pages   = {183--208}
++}
++
++@article{ihlanfeldt2023appraisal,
++  author  = {Ihlanfeldt, Keith and Rodgers, Luke P.},
++  title   = {Homestead Exemptions, Heterogeneous Assessment, and Property Tax Progressivity},
++  journal = {National Tax Journal},
++  year    = {2022},
++  volume  = {75},
++  number  = {1},
++  pages   = {7--31}
++}
++
++@article{baar1981property,
++  author  = {Baar, Kenneth K.},
++  title   = {Property Tax Assessment Discrimination Against Low-Income Neighborhoods},
++  journal = {The Urban Lawyer},
++  year    = {1981},
++  volume  = {13},
++  number  = {3},
++  pages   = {333--406}
++}
++
++@article{engle1975deassessment,
++  author  = {Engle, Robert F.},
++  title   = {De Facto Discrimination in Residential Assessments: {Boston}},
++  journal = {National Tax Journal},
++  year    = {1975},
++  volume  = {28},
++  number  = {4},
++  pages   = {445--451}
++}

added paper/sections/conclusion.tex +37 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Conclusion}
++\label{sec:concl}
++
++Matching more than half a million residential sales to the assessment
++rolls that tax them, this paper delivers the first province-wide audit of
++assessment equity in Canada. The verdict is unambiguous. Quebec's
++property-tax base is systematically regressive: within the same
++municipality, roll and year, the assessment ratio falls by 34\% per
++log-point of price in the descriptive fixed-effects estimate, and by 8\%
++under the most conservative measurement-error correction available.
++Ninety-five percent of municipalities fail the IAAO vertical-equity
++standard, ninety-nine percent fail its uniformity standard, and the median
++dwelling in the bottom local price decile pays roughly two thirds more
++property tax than uniform assessment would imply, while the top decile
++pays less. The inequity concentrates where mass appraisal is hardest ---
++old houses, plexes, land-heavy and luxury properties, thin markets ---
++doubles when triennial rolls fall behind a booming market, and disappears
++almost nowhere except Montr\'eal.
++
++These findings reframe the presumption that centralized, professionally
++regulated assessment systems are immune to the pathologies documented in
++the fragmented U.S. system. Quebec has uniform statutes, certified
++evaluators and statutory market-vintage synchronization --- and produces
++regressivity of U.S. magnitude anyway. The binding constraint is not
++governance but valuation technology and revaluation frequency, which is
++in fact encouraging: cycle length, pooled assessment services and
++vertical audit standards are policy variables, and inexpensive ones
++relative to the redistribution at stake.
++
++The matched sale--roll architecture built here supports a research agenda
++beyond this audit: linking assessment gaps to neighbourhood
++socio-demographics, exploiting the discrete arrival of new rolls to study
++price anchoring, and using the land--building decomposition to measure
++land-value dynamics at the parcel level. We pursue these in companion
++work.

added paper/sections/data.tex +101 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Data and sample construction}
++\label{sec:data}
++
++\subsection{Sources and matching}
++
++We start from a compiled registry of 745{,}119 residential-market
++transactions recorded in Quebec between January 2021 and July 2026, each
++carrying the sale price, sale date, civic address, geographic coordinates,
++and a listing-derived property type. Each transaction has been matched at
++the parcel level to the municipal assessment roll in force on the sale date.
++The match keys on geographic proximity between the transaction's coordinates
++and the roll unit's coordinates and is validated against the roll's recorded
++value; a composite score (maximum 220) summarizes the quality of the
++address, distance and value agreement. Matches are extremely tight: the
++median distance between the transaction and the matched roll unit is
++0.6~metres, and every observation retained in the estimation sample
++reproduces the roll's assessed value exactly.
++
++From the roll we observe the taxable value of the property
++($AV$, \texttt{valeur immeuble}), its land and building components, lot
++area, total floor area, year of construction, number of dwelling units, the
++standardized use code (CUBF), the roll vintage, and the statutory
++market-condition reference date discussed in Section~\ref{sec:inst}.
++
++\subsection{Sample restrictions}
++
++Table~\ref{tab:sumstats} describes the estimation sample; the selection
++cascade is as follows. We keep sales of residential use codes ---
++dwellings (CUBF 1000), cottages (1100), mobile homes (1211) and other
++residential (1990) --- which removes vacant land, commercial property and
++construction-in-progress (672{,}276 sales remain). We require a
++high-confidence roll match (distance $\le 50$~m and score $\ge 150$;
++568{,}711 sales), a positive assessed value and a price of at least
++\$50{,}000 --- the floor of the source registry, which also screens out
++most non-arm's-length transfers. Assessment ratios $r_i = AV_i/SP_i$ are
++trimmed at the 1st and 99th percentiles \emph{within each roll vintage}, so
++that the mechanical drift of ratio levels across vintages is not trimmed
++asymmetrically (557{,}325 sales). Finally, since all estimates compare
++sales within a municipality $\times$ roll $\times$ sale-year block
++(``cell''), we require at least 20 sales per cell. The estimation sample
++contains \textbf{522{,}769 sales} in 625 municipalities and 2{,}884 cells:
++345{,}053 single-family homes, 81{,}693 plexes (2--5 units), 81{,}461
++condominiums, 9{,}287 cottages and 4{,}701 mobile homes.
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{Summary statistics, estimation sample}
++\label{tab:sumstats}
++\small
++\input{../results/tables/summary_stats}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} 522{,}769 residential sales, January 2021 -- July
++2026, matched at the parcel level to the assessment roll in force at the
++sale date. The assessment ratio is assessed value divided by sale price.
++Roll lag is the number of months between the roll's statutory
++market-condition reference date (July 1) and the sale date. The assessed
++land share is the roll's land value divided by total assessed value.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\subsection{The raw pattern}
++
++The median assessment ratio is 0.78: the typical dwelling sells for about
++28\% more than its rolled value, the expected imprint of a rising market on
++back-dated rolls. Figure~\ref{fig:ratiodist} shows the distribution and its
++decomposition by roll lag: sales occurring within two years of the
++reference date centre near 0.87, while sales more than four years out
++centre near 0.64 --- the mechanical staleness gradient that our fixed
++effects absorb. Figure~\ref{fig:time} traces the same mechanics in
++calendar time: each roll vintage enters near parity with the market it was
++referenced on, then drifts down as prices rise, and the 2022--2023 rate
++shock is visible as a flattening of the drift. Everything that follows
++nets out this timing structure and asks a sharper question: \emph{within} a
++given municipality, roll, and year, do cheap and expensive homes face the
++same ratio?
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=\textwidth]{fig_ratio_dist.png}
++\caption{Assessment ratios. Panel A: distribution of $AV/SP$ across the
++estimation sample; the median is 0.78 and the dashed line marks parity.
++Panel B: kernel of the same distribution split by the number of months
++between the roll's market-condition reference date and the sale; older
++rolls sit systematically further below parity.}
++\label{fig:ratiodist}
++\end{figure}
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=\textwidth]{fig_time.png}
++\caption{Median assessment ratio by sale month and roll vintage. Each line
++follows sales assessed under one triennial vintage (labelled by entry
++year); monthly medians with fewer than 100 sales are suppressed. Vintages
++enter near their reference-date market level and drift down as prices
++rise; the flattening after 2022 reflects the interest-rate correction.}
++\label{fig:time}
++\end{figure}

added paper/sections/discussion.tex +81 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Discussion}
++\label{sec:disc}
++
++\subsection{Mechanisms}
++
++The evidence assembles into a coherent mechanism story with three layers.
++
++\paragraph{1. Hedonic difficulty, not assessor idiosyncrasy.} The
++heterogeneity profile tracks the difficulty of the valuation problem with
++almost embarrassing fidelity: condominiums (homogeneous, comparable-rich)
++are nearly equitable; old houses, plexes, cottages and high-land-share
++properties (heterogeneous, comparable-poor, unanchored by construction
++costs) are severely regressive. This is the pattern predicted if assessors
++run reasonable hedonic models whose unexplained component is largest ---
++and most value-correlated --- where the housing bundle is hardest to price
++\citep{amornsiripanitch2022residential, gloudemans2011fundamentals,
++bostic2007land}. Regression toward the mean in any imperfect valuation
++model over-values the cheap tail and under-values the expensive tail;
++our quantile profile, with proportionality holding at the bottom and
++collapsing at the top ($\beta(0.90) = 0.49$), shows Quebec's failure is
++concentrated where comparables are thinnest.
++
++\paragraph{2. Staleness amplifies the gap.} The doubling of the
++province-wide PRB in 2021--2022 shows that triennial rolls interact
++perversely with fast markets: when prices move 30--40\% between reference
++dates, the cross-sectional dispersion of appreciation --- which is itself
++correlated with price segment --- loads directly into measured inequity.
++Jurisdictions cannot control housing cycles, but they control revaluation
++frequency; annual-cycle systems mechanically cap this channel
++\citep{berry2021reassessing, ross2012assessor}.
++
++\paragraph{3. Scale helps: the Montr\'eal exception.} Montr\'eal ---
++the largest sales sample, a dedicated in-house assessment service, dense
++comparable markets in every segment --- is the only large jurisdiction
++with progressive assessments, in all six years. Mid-sized and small
++municipalities, many relying on contracted regional evaluators with
++limited modelling capacity, are uniformly regressive. The thin-market
++mechanism of \citet{mcmillen2008thin} operates within a single provincial
++rulebook: identical statutes, divergent outcomes, sorted by market depth
++and assessment resources. We note one caveat: Montr\'eal's measured
++progressivity may partly reflect within-city composition (its cell is the
++whole city, pooling boroughs), and borough-level analysis is a natural
++extension.
++
++\subsection{Policy implications}
++
++Three levers follow directly from the diagnosis, in increasing order of
++ambition. First, \textbf{shorten the cycle}: moving from triennial to
++annual (or indexed) rolls would eliminate the staleness component that
++dominated 2021--2022; several U.S. states and British Columbia already
++operate annual cycles. Second, \textbf{pool assessment capacity}: the
++regressivity gradient by municipality size argues for regional or
++provincial mass-appraisal services --- scale is the cheapest known
++technology for assessment quality. Third, \textbf{audit vertically, not
++just horizontally}: Quebec's oversight regime monitors median ratios and
++CODs but sets no explicit PRB requirement; adding the IAAO vertical
++standard to the ministry's audit criteria, with published
++municipality-level statistics like those in this paper, would make the
++equity dimension visible and contestable. Because low-priced-segment
++owners appeal least \citep{avenancio2022assessment, plummer2014evidence},
++supply-side correction --- better models --- will do more than
++demand-side remedies.
++
++\subsection{Limitations}
++
++Four limitations bound the interpretation. First, sale prices below
++\$50{,}000 are absent from the source registry and some residual
++non-arm's-length transfers may survive our screens; the robustness
++battery (dropping sub-\$100k sales, tightening trims) shows the
++conclusions survive, but the bottom-decile tax-shift magnitude should be
++read as an upper bound on that decile. Second, our data cover 2021--2026,
++an unusually turbulent market; the within-market elasticity is stable
++across the window, but levels statistics from calmer periods would differ.
++Third, we observe no appeals, renovations between assessment and sale, or
++conditions of sale; some of what we call horizontal noise is genuine
++unobserved quality change. Fourth, without owner demographics we cannot
++speak to the distributional incidence across income or racial groups as
++\citet{avenancio2022assessment} do --- linking these assessments to census
++tract characteristics is the obvious next step.

added paper/sections/institutional.tex +63 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Institutional setting: property assessment in Quebec}
++\label{sec:inst}
++
++Quebec's municipal fiscal regime is governed by the \textit{Loi sur la
++fiscalit\'e municipale} (LFM) and operationalized by the province's
++assessment manual, the \textit{Manuel d'\'evaluation fonci\`ere du
++Qu\'ebec} \citep{lfm2026, mamh2024manuel}. Four features matter for the
++measurement design of this paper.
++
++\paragraph{Triennial rolls.} Every municipality's assessment roll is
++redrawn on a fixed three-year cycle by a municipal body or a contracted
++private firm under the responsibility of a chartered evaluator. The roll
++lists, for each assessment unit, the taxable value of the property and its
++decomposition into land and building components, along with the structural
++descriptors used in valuation (lot area, floor area, year of construction,
++number of dwelling units, physical configuration, and a standardized use
++code, the CUBF).
++
++\paragraph{A statutory market-condition date.} Article 46 of the LFM
++requires that the values entered on a roll reflect the state of the market
++\emph{eighteen months before} the roll takes effect: a roll entering force
++on January~1 of year $t$ must value every property as of July~1 of year
++$t-2$. This single reference date is printed on the roll itself and is
++common to every property in the municipality. Two implications follow.
++First, in a rising market the \emph{level} of assessment ratios drifts
++mechanically below one as sales occur further from the reference date ---
++between 18 and 54 months elapse between the reference date and a sale, 36
++months at the median in our sample. This drift is a feature of the system,
++not an inequity, and our empirical design absorbs it entirely with
++municipality~$\times$~roll~$\times$~sale-year fixed effects. Second, and
++crucially, \emph{within} such a block every property shares the same market
++vintage, so cross-sectional patterns in ratios cannot be attributed to
++timing.
++
++\paragraph{Uniform taxation of the rolled value.} The municipal levy on a
++residential property is the rolled taxable value multiplied by the
++municipal rate (plus school taxes levied on the same base). There are no
++assessment-growth caps, homestead exemptions, or acquisition-value rules of
++the Californian or Floridian type: relative taxable values within a
++municipality equal relative assessed values. This makes the mapping from
++assessment error to tax-burden redistribution exact --- a property assessed
++10\% above the jurisdiction's median ratio pays 10\% more tax than uniform
++assessment would imply --- and motivates the tax-shift calculation of
++Section~\ref{sec:results}.
++
++\paragraph{Oversight and appeal.} The Ministry of Municipal Affairs (MAMH)
++prescribes methods and audits rolls; owners may request an administrative
++review and appeal to the Tribunal administratif du Qu\'ebec. Appeal rates
++for residential property are low. Unlike several U.S. jurisdictions
++studied in the literature \citep{plummer2014evidence,
++avenancio2022assessment}, Quebec's appeal system plays a minor quantitative
++role for the housing stock at large, which makes the valuation model
++itself --- rather than post-assessment litigation --- the natural locus of
++any inequity we measure.
++
++Taken together, these institutions imply that Quebec should be a
++\emph{best-case} environment for assessment equity: uniform provincial
++methodology, professional certification, statutory synchronization of
++market vintage, and a tax that consumes the rolled value without
++exemption-driven distortions. The magnitude of the inequity we document
++below should be read against this backdrop.

added paper/sections/introduction.tex +127 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Introduction}
++\label{sec:intro}
++
++The property tax is the fiscal backbone of Canadian local government: it
++finances roughly two thirds of municipal budgets and is, by statute, an
++\emph{ad valorem} tax --- every dwelling in a municipality is meant to be
++taxed in strict proportion to its market value. That proportionality rests
++entirely on the quality of the assessment roll. If assessors systematically
++overvalue inexpensive homes relative to expensive ones, the legal tax rate
++becomes a fiction: the effective tax rate falls with wealth, and the
++resulting redistribution --- from modest neighbourhoods toward affluent ones
++--- occurs silently, without any legislative decision, embedded in a
++technical document that few taxpayers ever contest. A rapidly growing
++literature documents exactly this pattern in the United States, where
++assessment regressivity has been measured at national scale
++\citep{berry2021reassessing, avenancio2022assessment,
++amornsiripanitch2022residential}. Whether the same pathology afflicts
++Canadian assessment systems --- centralized, professionally supervised, and
++widely presumed to be of high quality --- is essentially unknown.
++
++This paper provides the first province-wide answer for Quebec. We assemble
++522{,}769 residential transactions recorded between January 2021 and July
++2026 and match each sale, at the parcel level, to the triennial municipal
++assessment roll in force on the day of the sale. The match is performed on
++geographic coordinates and validated against the roll's recorded value; the
++median match distance is under one metre. The resulting file couples every
++sale price with the exact taxable value that generated the owner's property
++tax bill, together with the roll's decomposition of that value into land and
++building components, structural descriptors (lot area, floor area, year
++built, number of units), and the statutory market-condition reference date
++of the roll. The data span 625 municipalities --- from Montr\'eal
++(79{,}581 usable sales) to towns with barely a hundred --- and a housing
++cycle of unusual amplitude: the post-pandemic boom, the 2022--2023 interest
++rate shock, and the subsequent recovery.
++
++Quebec's institutional design makes it an unusually clean laboratory for
++equity measurement. Every municipality's roll is redone on a fixed
++three-year cycle, and by law (art.~46, \textit{Loi sur la fiscalit\'e
++municipale}) the values entered on a roll must reflect market conditions
++eighteen months before the roll takes effect --- a single, known reference
++date shared by every property in the municipality
++\citep{lfm2026, mamh2024manuel}. Within a municipality $\times$ roll
++$\times$ sale-year block, therefore, every assessment embodies the same
++market vintage, and any systematic relationship between assessment ratios
++and prices inside such a block is inequity, not timing. Our empirical design
++exploits exactly this structure: all regression estimates absorb 2{,}884
++municipality~$\times$~roll~$\times$~year fixed effects, so that vertical
++inequity is identified purely from comparisons between cheap and expensive
++dwellings facing the same assessor, the same roll, and the same market
++moment.
++
++Three findings emerge. First, assessment regressivity in Quebec is
++pervasive, large, and statistically unambiguous. The elasticity of the
++assessment ratio with respect to the sale price --- zero under proportional
++assessment --- is $-0.34$ (s.e.\ $0.03$) in the fixed-effects log-log
++regression. Because sale prices measure market value with noise, part of
++that estimate reflects mechanical attenuation \citep{kochin1982vertical,
++clapp1990new}; a rank-based instrumental-variables estimator in the spirit
++of \citet{clapp1990new} that purges this bias still yields $-0.08$
++(s.e.\ $0.02$). Under the IAAO ratio-study standards used by assessment
++authorities across North America \citep{iaao2013standard}, 95\% of Quebec
++municipalities fall outside the acceptable band for price-related bias and
++99\% exceed the uniformity (COD) ceiling. Strikingly, Montr\'eal --- the
++province's largest and best-resourced assessment jurisdiction --- is the
++\emph{only} major market whose assessments are progressive.
++
++Second, the inequity has a clear anatomy. Regressivity is roughly four times
++stronger for single-family homes and plexes than for condominiums, whose
++quasi-homogeneous units are easy to mass-appraise; it rises steeply with
++building age and with the assessed land share of the property, consistent
++with the difficulty of valuing land and depreciated structures
++\citep{bostic2007land, gloudemans2011fundamentals}; and it is worse in small
++municipalities, echoing the thin-market mechanism of \citet{mcmillen2008thin}.
++The quantile profile is equally telling: the log-log slope
++falls from $0.87$ at the first decile of the conditional value distribution
++to $0.49$ at the ninth, so the failure of proportionality is concentrated at
++the top --- expensive homes are not merely under-assessed, they are
++under-assessed at an accelerating rate.
++
++Third, the stakes are material. Within a taxing jurisdiction the levy is
++proportional to assessed value, so a property assessed above the
++jurisdiction median ratio pays exactly that percentage more tax than uniform
++assessment would imply. The median dwelling in the bottom within-market
++price decile pays roughly 65\% more property tax than under uniform
++assessment; the median dwelling in the top decile pays about 5\% less. Put
++differently, the effective tax schedule that Quebec's assessment machinery
++delivers is regressive enough to undo, within each municipality, a
++substantial share of whatever progressivity the rest of the fiscal system
++achieves --- a finding that complements \citet{carbonnier2024property}, who
++documents the regressivity of Quebec's property tax relative to income using
++survey data, but who could not observe the assessment channel isolated here.
++
++We also show \emph{when} the gap opens. Assessment ratios drift mechanically
++between reference dates, and the province-wide price-related bias doubles in
++2021--2022, precisely when the post-pandemic boom pulled market prices away
++from rolls anchored in 2018--2020 conditions --- direct evidence that
++infrequent revaluation is a first-order driver of measured inequity, as
++conjectured in the U.S. literature \citep{berry2021reassessing,
++ross2012assessor}. Because the diagnosis singles out stale rolls, thin
++markets, and hard-to-value property types --- rather than assessor
++discretion alone --- it maps directly into policy: shorter revaluation
++cycles, pooled assessment services for small municipalities, and targeted
++review of high-land-share and older properties.
++
++Beyond the Canadian evidence gap, the paper makes two methodological
++contributions to the ratio-study literature. It is, to our knowledge, the
++first large-scale equity study to exploit a statutory single-date reference
++regime to separate timing drift from genuine vertical inequity by design
++rather than by econometric correction; and it implements the full modern
++test battery --- IAAO diagnostics with bootstrap inference
++\citep{iaao2013standard}, the classical regression tests
++\citep{paglin1972equity, cheng1974property}, measurement-error-robust IV
++\citep{clapp1990new}, and quantile profiles \citep{mcmillen2020assessment}
++--- on more than half a million sales, allowing precise subgroup estimates
++that smaller samples cannot support.
++
++The remainder of the paper proceeds as follows. Section~\ref{sec:lit}
++positions the paper in the ratio-study and property-tax-equity literatures.
++Section~\ref{sec:inst} describes Quebec's assessment institutions.
++Section~\ref{sec:data} presents the data and the matched sample.
++Section~\ref{sec:method} lays out the diagnostics and econometric
++specifications. Section~\ref{sec:results} reports the results,
++Section~\ref{sec:robust} the robustness battery, Section~\ref{sec:disc}
++discusses mechanisms and policy implications, and Section~\ref{sec:concl}
++concludes.

added paper/sections/literature.tex +100 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Related literature}
++\label{sec:lit}
++
++\subsection{Measuring vertical inequity: from ratio studies to modern tests}
++
++The empirical study of assessment equity is nearly as old as the modern
++property tax. Its workhorse is the \emph{sales ratio study}: for each sale
++$i$, compute the ratio $r_i = AV_i/SP_i$ of assessed value to sale price and
++ask whether $r_i$ is uniform across the value distribution. The
++practitioner's canon --- codified in the IAAO \textit{Standard on Ratio
++Studies} \citep{iaao2013standard} and the mass-appraisal manuals of
++\citet{eckert1990property} and \citet{gloudemans2011fundamentals} ---
++summarizes a jurisdiction with the median ratio, the coefficient of
++dispersion (COD) for horizontal uniformity, and the price-related
++differential (PRD) or, since 2013, the coefficient of price-related bias
++(PRB) for vertical equity. These diagnostics discipline assessment practice
++across North America, and we report all of them, with bootstrap confidence
++intervals, for every Quebec municipality with sufficient sales.
++
++The academic literature grew out of dissatisfaction with those summary
++measures. \citet{paglin1972equity} proposed regressing assessed value on
++sale price in levels and reading regressivity off a positive intercept;
++\citet{cheng1974property} moved the test to logarithms, where the slope
++$\beta$ has the clean interpretation of an elasticity and $\beta<1$ signals
++regressivity. Subsequent work refined the specification menu ---
++\citet{bell1984administrative} and \citet{sunderman1990testing} allowed
++splines and structural breaks, and \citet{sirmans1995vertical} surveyed the
++proliferating test battery. A decisive critique came from
++\citet{kochin1982vertical} and \citet{kennedy1984unfair}: because the sale
++price is itself a noisy measure of market value, the log-log slope is
++attenuated toward zero even under perfectly proportional assessment ---
++ratio studies are biased \emph{toward} finding regressivity.
++\citet{clapp1990new} answered with a rank-based instrumental-variables
++estimator whose coarse instrument is nearly orthogonal to transitory price
++noise; \citet{goolsby1997assessment} and \citet{mcmillen2020assessment}
++document how severely the correction can matter, and
++\citet{quintos2020prb} extends the toolkit with machine-learning
++diagnostics. Our design follows this lesson to the letter: we report the
++na\"ive estimates, the Clapp IV, and quantile profiles
++\citep{mcmillen2020assessment} side by side, and treat the IV as the lower
++bound on true regressivity.
++
++\subsection{Evidence: pervasive regressivity, mostly documented in the U.S.}
++
++Substantively, the modern consensus is that assessment regressivity is the
++rule rather than the exception. \citet{berry2021reassessing} computes ratio
++statistics for essentially every U.S. county and finds that the
++lowest-decile home within a jurisdiction faces an effective tax rate roughly
++double that of the top decile; \citet{avenancio2022assessment} show in the
++\textit{QJE} that the same machinery generates a 10--13\% assessment gap
++against Black and Hispanic homeowners, driven partly by unpriced
++neighbourhood attributes and partly by differential appeals;
++\citet{amornsiripanitch2022residential} attributes much of the gradient to
++assessors' inability to capture unobserved quality that scales with price.
++Earlier strands documented the same pattern city by city ---
++\citet{engle1975deassessment} in Boston, \citet{baar1981property} across
++low-income neighbourhoods, \citet{mcmillen2011assessment} and
++\citet{mcmillen2008thin} in Illinois, where thin markets and infrequent
++reassessment emerge as structural drivers, \citet{hodge2017assessment} in
++collapsing Detroit, and \citet{plummer2014evidence} and
++\citet{ihlanfeldt2023appraisal} on the role of appeals and exemptions.
++\citet{ross2012assessor} shows regressivity responds to the institutional
++incentives of assessors themselves. The mechanism menu that emerges ---
++stale valuations, hard-to-value heterogeneous properties, thin markets,
++asymmetric appeals --- guides our heterogeneity analysis directly.
++
++\subsection{Canada: a conspicuous gap}
++
++For Canada the cupboard is nearly bare. Ontario's assessment corporation
++publishes an internal vertical-equity review of its own values
++\citep{mpac2022vertical}, and \citet{found2017property} compare tax
++\emph{rates} across cities, but neither measures sale-level equity.
++For Quebec, \citet{carbonnier2024property} establishes with survey microdata
++that property tax payments are steeply regressive relative to
++\emph{income}; the assessment channel --- whether the tax base itself is
++mismeasured against \emph{market value} --- has never been examined at
++scale, for Quebec or for any Canadian province. This is the gap the present
++paper fills. The Canadian case is of independent interest precisely because
++its institutions differ from the fragmented U.S. county system on the
++dimensions the U.S. literature blames: Quebec assessment is governed by a
++uniform provincial statute and manual \citep{lfm2026, mamh2024manuel},
++performed by certified professional evaluators, and synchronized to a
++statutory market-condition date. Finding large regressivity \emph{despite}
++this apparatus sharpens the interpretation considerably: the problem is not
++lax governance but the intrinsic difficulty of mass appraisal, compounded by
++triennial staleness.
++
++\subsection{Hedonic foundations}
++
++Finally, the paper connects to the hedonic tradition. Mass appraisal is
++applied hedonics \citep{rosen1974hedonic, malpezzi2003hedonic}: the
++assessor's model prices a bundle of structural and locational attributes,
++and equity failures are hedonic specification failures --- unpriced
++land-share gradients \citep{bostic2007land}, depreciation curvature, or
++neighbourhood effects that Quebec practitioners have long modelled in the
++Quebec City market \citep{desrosiers2000hedonic}. Reading our subgroup
++estimates through this lens turns an audit into a diagnosis: the assessment
++gap is largest exactly where the hedonic problem is hardest.

added paper/sections/methodology.tex +108 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Methodology}
++\label{sec:method}
++
++Let $AV_i$ denote the assessed value of property $i$ on the roll in force
++at its sale, $SP_i$ its sale price, and $r_i = AV_i/SP_i$ the assessment
++ratio. Under perfectly proportional (vertically equitable) assessment,
++$\mathbb{E}[r_i \mid SP_i]$ is constant. All tests below ask, in different
++ways, whether $r_i$ instead declines with value.
++
++\subsection{IAAO ratio-study diagnostics}
++
++For any group of sales we report the four statistics of the IAAO
++\textit{Standard on Ratio Studies} \citep{iaao2013standard}. The
++\emph{median ratio} measures the assessment level. The \emph{coefficient of
++dispersion},
++$\mathrm{COD} = 100 \cdot \operatorname{mean}\!\left(|r_i -
++\tilde r|\right) / \tilde r$ with $\tilde r$ the median ratio, measures
++horizontal uniformity; the standard deems residential CODs above 15
++unacceptable. The \emph{price-related differential},
++$\mathrm{PRD} = \bar r \,/\, (\sum_i AV_i / \sum_i SP_i)$, compares the
++unweighted and value-weighted mean ratios; PRD~$>1.03$ indicates
++regressivity. Because the PRD is sensitive to outliers, the standard's
++preferred vertical measure is the \emph{coefficient of price-related bias}
++(PRB), the slope $b$ in
++\begin{equation}
++\frac{r_i - \tilde r}{\tilde r}
++  \;=\; a + b \cdot \log_2\!\Big(\tfrac{1}{2}SP_i +
++  \tfrac{1}{2}\,AV_i/\tilde r\Big) + u_i ,
++\label{eq:prb}
++\end{equation}
++which measures the proportional change in the ratio per doubling of value;
++the acceptable band is $[-0.05, 0.05]$. We compute percentile-bootstrap
++confidence intervals for all four statistics. Because ratio levels drift
++with roll staleness, municipality-level diagnostics are computed within
++municipality~$\times$~sale-year blocks --- inside which a single roll is in
++force --- and aggregated across years by the median.
++
++\subsection{Regression tests with market-timing fixed effects}
++
++Our workhorse is the log-log specification of \citet{cheng1974property},
++\begin{equation}
++\ln AV_i \;=\; \alpha_{c(i)} + \beta \, \ln SP_i + \varepsilon_i ,
++\label{eq:cheng}
++\end{equation}
++where $c(i)$ indexes the municipality~$\times$~roll~$\times$~sale-year cell
++of sale $i$ and the $\alpha_c$ are 2{,}884 absorbed fixed effects.
++Proportionality implies $\beta = 1$; we report
++$\gamma \equiv \beta - 1$, the elasticity of the assessment \emph{ratio}
++with respect to price ($\gamma<0$: regressive). The fixed effects guarantee
++that $\gamma$ is identified only from comparisons of dwellings assessed by
++the same authority, on the same roll vintage, and sold in the same year ---
++purging the staleness drift, municipal composition, and aggregate market
++movements in one stroke. For completeness we also report the pooled
++regression without fixed effects and the levels test of
++\citet{paglin1972equity}. Inference is clustered at the municipality level
++throughout (625 clusters).
++
++\subsection{Measurement error and the Clapp instrument}
++
++Sale prices measure market value with idiosyncratic noise --- bilateral
++bargaining, unobserved conditions of sale --- so OLS on
++equation~\eqref{eq:cheng} suffers attenuation bias: $\hat\beta < 1$ even
++under proportional assessment \citep{kochin1982vertical, kennedy1984unfair}.
++Following \citet{clapp1990new}, we instrument $\ln SP_i$ with a coarse rank
++variable $Z_i \in \{-1, 0, +1\}$ that flags sales in the bottom or top
++third of \emph{both} the within-cell $\ln AV$ and $\ln SP$ distributions.
++Because $Z_i$ retains only ordinal information agreed on by both measures of
++value, it is (nearly) orthogonal to the transitory component of either, and
++the resulting two-stage least-squares estimate of $\beta$ is consistent
++under classical measurement error. The demeaned-within-cell implementation
++preserves the fixed-effects structure. Throughout the paper we treat
++$\gamma_{\text{IV}}$ as a conservative lower bound on regressivity and
++$\gamma_{\text{FE}}$ as the descriptive upper bound; the truth lies between.
++
++\subsection{Quantile profile and heterogeneity}
++
++Averaging can hide where proportionality fails. We estimate
++equation~\eqref{eq:cheng} by quantile regression on within-cell demeaned
++variables at $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$
++\citep{mcmillen2020assessment}, tracing $\beta(\tau)$ across the
++conditional distribution of assessed values. We then re-estimate the
++fixed-effects model on subsamples --- property class, building-age bands,
++assessed-land-share bands, roll-lag bands, municipality size, and sale year
++--- keeping only cells that retain at least 20 sales, to locate the
++inequity where the mass-appraisal problem is hardest.
++
++\subsection{Horizontal inequity}
++
++Vertical tests concern the \emph{mean} of $r_i$ given value; horizontal
++equity concerns its \emph{dispersion} among comparable properties
++\citep{allen2002measuring, sirmans2008vertical}. Beyond the COD, we ask
++who receives noisy assessments: we regress the absolute deviation of a
++sale's log ratio from its cell median,
++$|\ln r_i - \operatorname{med}_{c(i)} \ln r|$, on property characteristics
++(age, land share, property-class indicators) with cell fixed effects.
++
++\subsection{The implied tax shift}
++
++Because Quebec taxes the rolled value without exemptions or caps
++(Section~\ref{sec:inst}), a property whose ratio exceeds its jurisdiction's
++median by $x\%$ pays exactly $x\%$ more tax than uniform assessment would
++imply. For each sale we compute the relative assessment error
++$e_i = r_i / \operatorname{med}_{c(i)}(r) - 1$ and average it by
++within-cell sale-price decile. This translates the econometrics into the
++policy-relevant object: the percentage over- or under-payment of property
++tax by position in the local price distribution.

added paper/sections/results.tex +298 −0

++% Author: Simon-Pierre Boucher — contact@spboucher.ai
++% ============================================================================
++\section{Results}
++\label{sec:results}
++
++\subsection{Province-wide diagnostics}
++
++Table~\ref{tab:iaao} reports the IAAO statistics for the full sample and by
++sale year. Three facts stand out. First, uniformity is poor everywhere: the
++province-wide COD is 28.5, and even within municipality-year blocks the
++median municipality posts a COD of 26 --- nearly twice the IAAO ceiling of
++15 for residential property; 99\% of municipalities exceed it
++(Figure~\ref{fig:cod}). Second, every vertical measure points the same way:
++the PRD is 1.072 (acceptable range 0.98--1.03) and the PRB is $-0.029$ with
++a bootstrap confidence interval far from zero --- ratios fall by about 3\%
++of the median with every doubling of value. Third, the time pattern is
++diagnostic: the PRB collapses to $-0.104$ and $-0.117$ in 2021 and 2022 ---
++when the boom pulled prices away from rolls referenced on 2018--2020
++markets --- and shrinks to $-0.002$ in 2024 before widening again. Vertical
++inequity is not a fixed institutional constant; it breathes with the gap
++between the market and the roll's vintage.
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{IAAO ratio-study diagnostics, province-wide}
++\label{tab:iaao}
++\small
++\input{../results/tables/iaao}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Median assessment ratio, coefficient of dispersion
++(COD), price-related differential (PRD) and coefficient of price-related
++bias (PRB), computed on the estimation sample; 95\% percentile-bootstrap
++confidence intervals (200 replications) in brackets. IAAO acceptable
++ranges: COD $\le 15$ (single-family residential), PRD $\in [0.98, 1.03]$,
++PRB $\in [-0.05, 0.05]$.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=\textwidth]{fig_cod.png}
++\caption{Horizontal uniformity. Panel A: distribution of municipality-level
++CODs (median across sale-year blocks); the IAAO ceiling for residential
++property is 15. Panel B: COD against the number of sales in the
++municipality; dispersion is worst in the thinnest markets.}
++\label{fig:cod}
++\end{figure}
++
++Table~\ref{tab:cities} disaggregates the ten largest markets.
++Regressivity is significant in nine of them --- Sherbrooke ($-0.39$),
++Trois-Rivi\`eres ($-0.32$) and Terrebonne ($-0.29$) are the worst --- in
++every single sale year. The exception is Montr\'eal, whose PRB of $+0.07$
++makes it the lone \emph{progressive} large jurisdiction in the province, in
++all six years. We return to this contrast in Section~\ref{sec:disc}.
++Figure~\ref{fig:map} maps the municipal estimates: regressive (red)
++municipalities carpet the province, with the Montr\'eal agglomeration a
++solitary blue island, and Figure~\ref{fig:prbmuni} shows the distribution:
++99\% of the 261 municipalities with at least 100 usable sales have a
++negative PRB, and 95\% fall below the IAAO band.
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{Assessment equity in the ten largest markets}
++\label{tab:cities}
++\small
++\input{../results/tables/iaao_cities}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Statistics computed within municipality
++$\times$ sale-year blocks (a single roll in force per block) and
++aggregated across years by the median; $n$ is the total number of sales.
++The last column is the share of the six sale years in which the
++municipality's PRB is negative.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=0.72\textwidth]{fig_prb_muni.png}
++\caption{Distribution of municipality-level PRB across the 261
++municipalities with at least 100 usable sales. The shaded band is the IAAO
++acceptable range $[-0.05, 0.05]$.}
++\label{fig:prbmuni}
++\end{figure}
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=\textwidth]{fig_map.png}
++\caption{The geography of vertical inequity. Municipality-level PRB
++(median across sale-year blocks); bubble area is proportional to the
++square root of the number of sales. Montr\'eal is the only large
++progressive jurisdiction.}
++\label{fig:map}
++\end{figure}
++
++\subsection{Regression estimates}
++
++Table~\ref{tab:vertical} presents the econometric core.
++Figure~\ref{fig:binscatter} shows the raw material: within
++municipality~$\times$~roll~$\times$~year cells, mean log assessment ratios
++fall monotonically --- and remarkably linearly through the central 90\% of
++the distribution --- as log prices rise. The pooled \citet{cheng1974property}
++elasticity is $\beta = 0.812$ (s.e.\ 0.029); absorbing the 2{,}884 cell
++fixed effects \emph{steepens} it to $\beta = 0.656$, i.e.\
++$\gamma_{\text{FE}} = -0.344$: within the same market and vintage, a home
++priced 10\% above its neighbours is assessed only about 6.6\% higher, so
++its ratio is 3.4\% lower. The levels test of \citet{paglin1972equity}
++agrees: the intercept is \$31{,}729 (s.e.\ \$5{,}378) --- inexpensive
++properties carry a fixed assessment premium.
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{Vertical-inequity regressions}
++\label{tab:vertical}
++\small
++\input{../results/tables/vertical}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Dependent variable: $\ln AV$. Column (1): pooled
++OLS. Column (2): absorbing least squares with 2{,}884 municipality
++$\times$ roll $\times$ sale-year fixed effects. Column (3): within-cell
++2SLS using the \citet{clapp1990new} rank instrument
++$Z \in \{-1,0,+1\}$. Standard errors clustered by municipality (625
++clusters) in parentheses. Stars test $H_0\!: \beta = 1$ (proportional
++assessment); *** $p<0.01$.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=0.72\textwidth]{fig_binscatter.png}
++\caption{Vertical inequity within markets. Mean of the within-cell demeaned
++log assessment ratio by vigintile of the within-cell demeaned log sale
++price; whiskers are 95\% confidence intervals (they are smaller than the
++markers). The fitted line is the sale-weighted least-squares slope.}
++\label{fig:binscatter}
++\end{figure}
++
++How much of this is measurement-error artefact? The \citet{clapp1990new}
++instrument delivers $\beta_{\text{IV}} = 0.917$ (s.e.\ 0.021):
++attenuation indeed accounts for the majority of the na\"ive gap, exactly
++as the critique of \citet{kochin1982vertical} predicts --- but the
++corrected elasticity remains four standard errors below one. Genuine
++regressivity of roughly $-0.08$ survives the harshest correction in the
++literature. An 8\% ratio penalty per log-point of value is economically
++large: across the interquartile price range within a typical market
++(roughly 0.8 log points), it generates a 6--7\% differential in effective
++tax rates from vertical inequity alone, before adding the much larger
++dispersion documented below.
++
++\subsection{Where proportionality fails: the quantile profile}
++
++Figure~\ref{fig:quantile} plots $\beta(\tau)$. The profile is strikingly
++asymmetric: $\beta$ is 0.87 at $\tau=0.10$ and 0.86 at $\tau=0.25$, then
++falls to 0.70 at $\tau=0.75$ and 0.49 at $\tau=0.90$
++(Table~\ref{tab:quantile} in the robustness section reports the
++coefficients). Assessment tracks the market tolerably well through the
++middle of the distribution and loses contact at the top: the most expensive
++homes within each market are under-assessed at an accelerating rate. This
++is the signature of hedonic truncation --- mass-appraisal models built on
++mid-market comparables extrapolate poorly into thin luxury segments
++\citep{amornsiripanitch2022residential, gloudemans2011fundamentals}.
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=0.72\textwidth]{fig_quantile.png}
++\caption{Quantile profile of the Cheng elasticity. Quantile regressions of
++within-cell demeaned $\ln AV$ on demeaned $\ln SP$; the shaded band is the
++pointwise 95\% confidence interval. Reference lines mark proportionality
++($\beta=1$) and the FE and IV point estimates.}
++\label{fig:quantile}
++\end{figure}
++
++\subsection{The anatomy of regressivity}
++
++Figure~\ref{fig:het} and Table~\ref{tab:het} dissect
++$\gamma_{\text{FE}}$ by subsample, and the pattern reads like a map of
++mass-appraisal difficulty:
++
++\begin{itemize}
++\item \textbf{Property class.} Condominiums --- homogeneous units with
++abundant comparables --- show $\gamma = -0.12$, single-family homes
++$-0.45$ and plexes $-0.49$. The assessor's problem, not the assessor's
++jurisdiction, drives the gradient.
++\item \textbf{Building age.} Regressivity deepens from $-0.30$ (under 20
++years) to $-0.43$ (over 60): depreciation and renovation heterogeneity are
++hard to observe from the roll.
++\item \textbf{Land share.} The starkest gradient: $-0.26$ where land is
++under 20\% of assessed value versus $-0.50$ where it exceeds 40\%.
++Land is the component without construction-cost anchoring
++\citep{bostic2007land}, and properties whose value is mostly land are
++assessed worst.
++\item \textbf{Municipality size.} Large markets ($>$10k sales) do
++better ($-0.27$) than mid-sized ones ($-0.41$), consistent with
++\citet{mcmillen2008thin}.
++\item \textbf{Stability.} Estimates are remarkably stable across sale
++years ($-0.32$ to $-0.38$) and roll lags --- unlike the \emph{level}
++statistics, the within-market elasticity is a structural feature of the
++valuation technology, not of the cycle.
++\end{itemize}
++
++\begin{figure}[p]
++\centering
++\includegraphics[width=0.85\textwidth]{fig_heterogeneity.png}
++\caption{Heterogeneity of the within-market elasticity
++$\gamma$. Each point is a separate fixed-effects estimate of
++equation~\eqref{eq:cheng} on the indicated subsample; whiskers are 95\%
++confidence intervals with municipality-clustered standard errors.}
++\label{fig:het}
++\end{figure}
++
++\begin{table}[p]
++\centering
++\begin{threeparttable}
++\caption{Heterogeneity of vertical inequity}
++\label{tab:het}
++\small
++\input{../results/tables/heterogeneity}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Cheng fixed-effects estimates of
++$\gamma = \beta - 1$ by subsample; cells with fewer than 20 sales after
++masking are dropped. Municipality-clustered standard errors in
++parentheses. *, **, *** denote significance at 10\%, 5\%, 1\%.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\subsection{Horizontal inequity}
++
++Table~\ref{tab:horizontal} reports the dispersion regression. The mean
++absolute deviation of a sale's log ratio from its cell median is 0.188 ---
++the typical Quebec dwelling is assessed 19\% away from its market's norm,
++an enormous horizontal lottery consistent with the COD evidence. The
++deviation rises by 0.75~log points per decade of building age and by
++5.2~points for cottages, and falls by 5.5~points for condominiums.
++Old, idiosyncratic, land-heavy properties receive not only biased but
++\emph{noisy} valuations.
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{Horizontal inequity: who receives noisy assessments?}
++\label{tab:horizontal}
++\small
++\input{../results/tables/horizontal}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Dependent variable:
++$|\ln r_i - \operatorname{med}_{c(i)} \ln r|$. Absorbing least squares
++with cell fixed effects; municipality-clustered standard errors.
++Omitted class: single-family. *, **, *** denote significance at 10\%,
++5\%, 1\%.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}
++
++\subsection{The implied tax shift}
++
++Figure~\ref{fig:taxshift} and Table~\ref{tab:taxshift} translate the
++estimates into tax dollars. Within the median market, the median dwelling
++in the bottom price decile is assessed 65\% above the local norm --- and
++therefore pays 65\% more property tax than uniform assessment would imply.
++The overpayment falls to $+7\%$ in the second decile and crosses zero at
++the median; the top decile underpays by about 5\%. The redistribution is
++strongly convex: the burden of assessment error is overwhelmingly
++concentrated on the cheapest tenth of the housing stock --- precisely the
++segment where owners have the least capacity to appeal and the highest
++housing-cost burdens. The magnitudes rival those documented for the U.S.
++by \citet{berry2021reassessing}, in a system with none of the U.S.'s
++institutional fragmentation.
++
++\begin{figure}[t]
++\centering
++\includegraphics[width=0.85\textwidth]{fig_taxshift.png}
++\caption{The implied property-tax shift. Median relative assessment error
++$e_i = r_i/\operatorname{med}_{c(i)}(r) - 1$ by within-market sale-price
++decile. Because the levy is proportional to the rolled value, $e_i$ equals
++the percentage over- or under-payment relative to uniform assessment.}
++\label{fig:taxshift}
++\end{figure}
++
++\begin{table}[t]
++\centering
++\begin{threeparttable}
++\caption{Excess tax burden by within-market price decile}
++\label{tab:taxshift}
++\small
++\input{../results/tables/taxshift}
++\begin{tablenotes}[flushleft]\footnotesize
++\item \textit{Notes:} Relative assessment error by within-cell sale-price
++decile. The mean is sensitive to extreme low-price sales; the median is
++the preferred summary. Standard errors of the mean in parentheses.
++\end{tablenotes}
++\end{threeparttable}
++\end{table}

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spb/wp10_uqo

WP10 — The Assessment Gap in Quebec: full research project

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