% ============================================================================= % QWHPI — Quebec Weekly Housing Price Index % Author : Simon-Pierre Boucher % Contact : contact@spboucher.ai % File : paper/qwhpi.tex % Purpose : Methodology paper — A High-Frequency Hedonic Housing Price Index % for Quebec. % ============================================================================= \documentclass[11pt]{article} \usepackage[margin=1.1in]{geometry} \usepackage{amsmath, amssymb, booktabs, graphicx, natbib, hyperref} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \title{A High-Frequency Hedonic Housing Price Index for Quebec:\\ Weekly Measurement with Hierarchical Pooling} \author{Simon-Pierre Boucher\thanks{Contact: contact@spboucher.ai. Code and data pipeline: QWHPI platform repository.}} \date{August 2026} \begin{document} \maketitle \begin{abstract} We construct quality-adjusted \emph{weekly} housing price indexes for Quebec from roughly 745{,}000 geocoded residential transactions (2021--2026), published for the province, its 17 administrative regions, and ten major municipalities, by property type. The method combines a pooled hedonic regression with week-by-type time effects (stage~1) and a hierarchical state-space model that shrinks thin geography--type cells toward their parent trends (stage~2). A repeat-sales benchmark, a downsampling experiment, and a composition-shock simulation validate the design. Weekly measurement is feasible far below conventional liquidity thresholds: with only 25 transactions per week the smoothed path deviates from the full-sample path by under 0.8\% RMSE. All series carry calibrated uncertainty bands, reliability grades, and a real-time (one-sided) variant that never uses future data. \end{abstract} \section{Introduction} House price indexes are published monthly or quarterly almost everywhere, yet the underlying transactions arrive continuously. The obstacle to weekly measurement is not conceptual but statistical: weekly cells are thin, and naive weekly averages confound price change with composition change. The composition problem is classical \citep{court1939, rosen1974}; the thinness problem is acute at weekly frequency for any geography below a large metropolitan area. \citet{anenberg2017} demonstrate that timely house-price measurement is feasible with appropriate pooling; we push frequency to the week and geography to the municipality for a single Canadian province. Our contribution is a transparent two-stage architecture. Stage~1 is a single pooled hedonic regression over the full sample in the time-dummy tradition \citep{hill2013, handbook2013}: structural characteristics, fine location fixed effects, and week$\times$type effects that constitute the provincial weekly price paths. Stage~2 treats each sub-provincial cell's weekly deviation from its parent path as a latent random walk observed through the cell's weekly mean residual, estimated by Kalman filtering \citep{kalman1960, durbin2012}. The observation variance scales inversely with the week's transaction count, so liquid weeks speak for themselves while thin weeks shrink toward the parent trend --- an empirical-Bayes resolution of the thinness problem in the spirit of hierarchical modeling. \section{Data} The input is a registry extract of 745{,}119 residential transactions in 1{,}100+ Quebec municipalities from January 2021 through July 2026, with price, date, address, coordinates, property type (single-family, condominium, plex, undetermined), floor area, year built, building type, and the municipal assessment value. Coordinates are spatially joined to the official SDA 1/20{,}000 administrative boundaries (100\% within-polygon match; 99.9\% agreement with the raw city text field). The \emph{undetermined} type (14.4\% of records, 95\%+ missing on all structural characteristics) never enters any published index. Screens are economically motivated and fully tabulated: transfers priced outside $[0.2, 5.0]$ times the municipal assessment (predominantly non-arm's-length family transfers and estate settlements), conveyances above \$10M (portfolio scale), and duplicate registrations --- 3.5\% of rows in total, each flagged with a reason and never silently deleted. Implausible characteristics (floor area outside $[20, 2000]$~m$^2$, year built after the sale year) are nulled with indicators, not dropped. The research sample is 626{,}127 transactions across all 291 weeks. \section{Method} \subsection{Stage 1: pooled hedonic} For transaction $i$ in week $t(i)$, type $k(i)$, location $\ell(i)$: \begin{equation} \log P_i = \beta_1 \log \mathrm{FA}_i + \beta_2 \mathbb{1}[\mathrm{FA\ miss}]_i + f(\mathrm{age}_i) + \gamma_{b(i)} + \alpha_{\ell(i)} + \delta_{t(i),k(i)} + \varepsilon_i , \end{equation} with age bins $f(\cdot)$, building-type effects $\gamma$, forward-sortation-area fixed effects $\alpha$ (the postal-code level; finer than municipality), and week$\times$type effects $\delta_{t,k}$. Missing floor areas are imputed by conditional medians within type$\times$municipality and always flagged; no future price information enters any imputation. The assessment value is excluded from the headline model to avoid valuation feedback (it appears only in a robustness specification and in the screens). Estimation is by high-dimensional fixed-effects least squares on the full pooled sample ($R^2 = 0.60$); the paths $\delta_{\cdot,k}$, rebased so the 2021 average equals 100, are the provincial indexes per type. Independent weekly regressions are never run. \subsection{Stage 2: hierarchical weekly state} Let $e_i = \log P_i - \widehat{\log P_i}$ be stage-1 residuals. For a cell $c$ (region$\times$type, then municipality$\times$type) define the weekly mean $\bar e_{c,t}$ over $n_{c,t}$ transactions. The latent deviation $d_{c,t}$ of the cell from its parent's path evolves as \begin{equation} \bar e_{c,t} = d_{c,t} + \epsilon_{c,t}, \quad \epsilon_{c,t} \sim N(0, \sigma^2_c / n_{c,t}), \qquad d_{c,t} = d_{c,t-1} + \eta_{c,t}, \quad \eta_{c,t} \sim N(0, \tau^2_c), \end{equation} with $\sigma^2_c$ the cell's residual variance and $\tau^2_c$ estimated by maximum likelihood via the prediction-error decomposition. Zero-transaction weeks are pure predictions --- the weekly grid is never broken. Municipality deviations are estimated relative to the smoothed regional path, so the published municipal index is $\delta_{t,k} + \hat d^{\,\mathrm{region}}_{t} + \hat d^{\,\mathrm{city}}_{t}$. The Kalman gain provides an interpretable shrinkage weight; the posterior variance combines with the delta-method variance of $\delta_{t,k}$ into the published 95\% band. Two variants ship for every series: the raw weekly estimate and a strictly one-sided filtered path (\emph{no lookahead}) for real-time use; a two-sided smoother is retained for research. The ``all types'' series is a fixed-weight combination of the three type indexes with full-sample transaction shares, in the stratification tradition of \citet{handbook2013}. \section{Validation} \paragraph{Repeat sales.} A Bailey--Muth--Nourse index \citep{bailey1963, caseshiller1989} on same-unit pairs (unit signature = address, floor area, year built; 90-day minimum gap; 63k+ pairs) tracks the hedonic indexes' relative paths closely: Montréal condominiums end 8.8\% below the provincial condominium index by repeat sales versus 8.2\% by the hierarchical model. Hedonic levels sit a few percent above repeat sales, consistent with the well-documented depreciation/renovation bias of repeat-sales measures \citep{goetzmann1992, bourassa2006}. Notably, a city-specific time-dummy regression \emph{overstates} Montréal's divergence ($-14\%$) --- pooled coefficients with local deviations are closer to the composition-free benchmark than fully local estimation. \paragraph{Downsampling.} Using Montréal condominiums ($\sim$145/week) as a laboratory, we thin each week to a target count and compare against the full-sample path (20 replications): RMSE rises from 0.32\% (100/wk) to 0.64\% (50/wk), 0.80\% (25/wk), 1.15\% (15/wk), 1.36\% (10/wk) and 1.63\% (5/wk); 95\% CI coverage stays above 0.97 down to 25/week and degrades to $\sim$0.83 at 10/week. These results anchor the published A--E reliability grades and the decision to publish (with warnings) rather than suppress thin cells. \paragraph{Composition shock.} Removing 70\% of below-median floor-area single-family sales for 26 weeks moves the raw median by $+11.7\%$ and the hedonic index by $-0.7\%$: the index measures prices, not mix. \paragraph{Seasonality.} Annual harmonics on weekly log-changes are jointly insignificant for all tested series ($p > 0.36$); all series are therefore published NSA. \section{Findings} Between the 2021 base and July 2026 the provincial all-type index rises to $\sim$156 ($+56\%$), with the 2022 spring boom, the 2022--23 rate-hike correction, and a sustained 2024--26 expansion clearly timed at weekly resolution. Regional heterogeneity is large: Montréal condominiums appreciate roughly 27\% while several regions exceed 70\%. The weekly index dates turning points 4--8 weeks earlier than the same methodology aggregated to monthly frequency. The assessment-gap module (sale price / municipal assessment) tracks the mechanical lag of triennial assessment rolls and is published as a separate concept, never mixed with the price index. \section{Conclusion} Weekly, quality-adjusted, sub-provincial price measurement is practical with transparent econometrics: one pooled hedonic regression plus one small state-space model per cell, running end-to-end in under a minute on a laptop. Honest uncertainty --- confidence bands, reliability grades, effective sample sizes, partial-week flags, and preserved vintages --- is what makes publication of thin cells defensible. \bibliographystyle{apalike} \bibliography{references} \end{document}