% 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}