% =============================================================================
% 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.
% =============================================================================
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\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.

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