% 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}
