spb/wp10_uqo Public

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Results}4\label{sec:results}56\subsection{Province-wide diagnostics}78Table~\ref{tab:iaao} reports the IAAO statistics for the full sample and by9sale year. Three facts stand out. First, uniformity is poor everywhere: the10province-wide COD is 28.5, and even within municipality-year blocks the11median municipality posts a COD of 26 --- nearly twice the IAAO ceiling of1215 for residential property; 99\% of municipalities exceed it13(Figure~\ref{fig:cod}). Second, every vertical measure points the same way:14the PRD is 1.072 (acceptable range 0.98--1.03) and the PRB is $-0.029$ with15a bootstrap confidence interval far from zero --- ratios fall by about 3\%16of the median with every doubling of value. Third, the time pattern is17diagnostic: the PRB collapses to $-0.104$ and $-0.117$ in 2021 and 2022 ---18when the boom pulled prices away from rolls referenced on 2018--202019markets --- and shrinks to $-0.002$ in 2024 before widening again. Vertical20inequity is not a fixed institutional constant; it breathes with the gap21between the market and the roll's vintage.2223\begin{table}[t]24\centering25\begin{threeparttable}26\caption{IAAO ratio-study diagnostics, province-wide}27\label{tab:iaao}28\small29\input{../results/tables/iaao}30\begin{tablenotes}[flushleft]\footnotesize31\item \textit{Notes:} Median assessment ratio, coefficient of dispersion32(COD), price-related differential (PRD) and coefficient of price-related33bias (PRB), computed on the estimation sample; 95\% percentile-bootstrap34confidence intervals (200 replications) in brackets. IAAO acceptable35ranges: COD $\le 15$ (single-family residential), PRD $\in [0.98, 1.03]$,36PRB $\in [-0.05, 0.05]$.37\end{tablenotes}38\end{threeparttable}39\end{table}4041\begin{figure}[t]42\centering43\includegraphics[width=\textwidth]{fig_cod.png}44\caption{Horizontal uniformity. Panel A: distribution of municipality-level45CODs (median across sale-year blocks); the IAAO ceiling for residential46property is 15. Panel B: COD against the number of sales in the47municipality; dispersion is worst in the thinnest markets.}48\label{fig:cod}49\end{figure}5051Table~\ref{tab:cities} disaggregates the ten largest markets.52Regressivity is significant in nine of them --- Sherbrooke ($-0.39$),53Trois-Rivi\`eres ($-0.32$) and Terrebonne ($-0.29$) are the worst --- in54every single sale year. The exception is Montr\'eal, whose PRB of $+0.07$55makes it the lone \emph{progressive} large jurisdiction in the province, in56all six years. We return to this contrast in Section~\ref{sec:disc}.57Figure~\ref{fig:map} maps the municipal estimates: regressive (red)58municipalities carpet the province, with the Montr\'eal agglomeration a59solitary blue island, and Figure~\ref{fig:prbmuni} shows the distribution:6099\% of the 261 municipalities with at least 100 usable sales have a61negative PRB, and 95\% fall below the IAAO band.6263\begin{table}[t]64\centering65\begin{threeparttable}66\caption{Assessment equity in the ten largest markets}67\label{tab:cities}68\small69\input{../results/tables/iaao_cities}70\begin{tablenotes}[flushleft]\footnotesize71\item \textit{Notes:} Statistics computed within municipality72$\times$ sale-year blocks (a single roll in force per block) and73aggregated across years by the median; $n$ is the total number of sales.74The last column is the share of the six sale years in which the75municipality's PRB is negative.76\end{tablenotes}77\end{threeparttable}78\end{table}7980\begin{figure}[t]81\centering82\includegraphics[width=0.72\textwidth]{fig_prb_muni.png}83\caption{Distribution of municipality-level PRB across the 26184municipalities with at least 100 usable sales. The shaded band is the IAAO85acceptable range $[-0.05, 0.05]$.}86\label{fig:prbmuni}87\end{figure}8889\begin{figure}[t]90\centering91\includegraphics[width=\textwidth]{fig_map.png}92\caption{The geography of vertical inequity. Municipality-level PRB93(median across sale-year blocks); bubble area is proportional to the94square root of the number of sales. Montr\'eal is the only large95progressive jurisdiction.}96\label{fig:map}97\end{figure}9899\subsection{Regression estimates}100101Table~\ref{tab:vertical} presents the econometric core.102Figure~\ref{fig:binscatter} shows the raw material: within103municipality~$\times$~roll~$\times$~year cells, mean log assessment ratios104fall monotonically --- and remarkably linearly through the central 90\% of105the distribution --- as log prices rise. The pooled \citet{cheng1974property}106elasticity is $\beta = 0.812$ (s.e.\ 0.029); absorbing the 2{,}884 cell107fixed effects \emph{steepens} it to $\beta = 0.656$, i.e.\108$\gamma_{\text{FE}} = -0.344$: within the same market and vintage, a home109priced 10\% above its neighbours is assessed only about 6.6\% higher, so110its ratio is 3.4\% lower. The levels test of \citet{paglin1972equity}111agrees: the intercept is \$31{,}729 (s.e.\ \$5{,}378) --- inexpensive112properties carry a fixed assessment premium.113114\begin{table}[t]115\centering116\begin{threeparttable}117\caption{Vertical-inequity regressions}118\label{tab:vertical}119\small120\input{../results/tables/vertical}121\begin{tablenotes}[flushleft]\footnotesize122\item \textit{Notes:} Dependent variable: $\ln AV$. Column (1): pooled123OLS. Column (2): absorbing least squares with 2{,}884 municipality124$\times$ roll $\times$ sale-year fixed effects. Column (3): within-cell1252SLS using the \citet{clapp1990new} rank instrument126$Z \in \{-1,0,+1\}$. Standard errors clustered by municipality (625127clusters) in parentheses. Stars test $H_0\!: \beta = 1$ (proportional128assessment); *** $p<0.01$.129\end{tablenotes}130\end{threeparttable}131\end{table}132133\begin{figure}[t]134\centering135\includegraphics[width=0.72\textwidth]{fig_binscatter.png}136\caption{Vertical inequity within markets. Mean of the within-cell demeaned137log assessment ratio by vigintile of the within-cell demeaned log sale138price; whiskers are 95\% confidence intervals (they are smaller than the139markers). The fitted line is the sale-weighted least-squares slope.}140\label{fig:binscatter}141\end{figure}142143How much of this is measurement-error artefact? The \citet{clapp1990new}144instrument delivers $\beta_{\text{IV}} = 0.917$ (s.e.\ 0.021):145attenuation indeed accounts for the majority of the na\"ive gap, exactly146as the critique of \citet{kochin1982vertical} predicts --- but the147corrected elasticity remains four standard errors below one. Genuine148regressivity of roughly $-0.08$ survives the harshest correction in the149literature. An 8\% ratio penalty per log-point of value is economically150large: across the interquartile price range within a typical market151(roughly 0.8 log points), it generates a 6--7\% differential in effective152tax rates from vertical inequity alone, before adding the much larger153dispersion documented below.154155\subsection{Where proportionality fails: the quantile profile}156157Figure~\ref{fig:quantile} plots $\beta(\tau)$. The profile is strikingly158asymmetric: $\beta$ is 0.87 at $\tau=0.10$ and 0.86 at $\tau=0.25$, then159falls to 0.70 at $\tau=0.75$ and 0.49 at $\tau=0.90$160(Table~\ref{tab:quantile} in the robustness section reports the161coefficients). Assessment tracks the market tolerably well through the162middle of the distribution and loses contact at the top: the most expensive163homes within each market are under-assessed at an accelerating rate. This164is the signature of hedonic truncation --- mass-appraisal models built on165mid-market comparables extrapolate poorly into thin luxury segments166\citep{amornsiripanitch2022residential, gloudemans2011fundamentals}.167168\begin{figure}[t]169\centering170\includegraphics[width=0.72\textwidth]{fig_quantile.png}171\caption{Quantile profile of the Cheng elasticity. Quantile regressions of172within-cell demeaned $\ln AV$ on demeaned $\ln SP$; the shaded band is the173pointwise 95\% confidence interval. Reference lines mark proportionality174($\beta=1$) and the FE and IV point estimates.}175\label{fig:quantile}176\end{figure}177178\subsection{The anatomy of regressivity}179180Figure~\ref{fig:het} and Table~\ref{tab:het} dissect181$\gamma_{\text{FE}}$ by subsample, and the pattern reads like a map of182mass-appraisal difficulty:183184\begin{itemize}185\item \textbf{Property class.} Condominiums --- homogeneous units with186abundant comparables --- show $\gamma = -0.12$, single-family homes187$-0.45$ and plexes $-0.49$. The assessor's problem, not the assessor's188jurisdiction, drives the gradient.189\item \textbf{Building age.} Regressivity deepens from $-0.30$ (under 20190years) to $-0.43$ (over 60): depreciation and renovation heterogeneity are191hard to observe from the roll.192\item \textbf{Land share.} The starkest gradient: $-0.26$ where land is193under 20\% of assessed value versus $-0.50$ where it exceeds 40\%.194Land is the component without construction-cost anchoring195\citep{bostic2007land}, and properties whose value is mostly land are196assessed worst.197\item \textbf{Municipality size.} Large markets ($>$10k sales) do198better ($-0.27$) than mid-sized ones ($-0.41$), consistent with199\citet{mcmillen2008thin}.200\item \textbf{Stability.} Estimates are remarkably stable across sale201years ($-0.32$ to $-0.38$) and roll lags --- unlike the \emph{level}202statistics, the within-market elasticity is a structural feature of the203valuation technology, not of the cycle.204\end{itemize}205206\begin{figure}[p]207\centering208\includegraphics[width=0.85\textwidth]{fig_heterogeneity.png}209\caption{Heterogeneity of the within-market elasticity210$\gamma$. Each point is a separate fixed-effects estimate of211equation~\eqref{eq:cheng} on the indicated subsample; whiskers are 95\%212confidence intervals with municipality-clustered standard errors.}213\label{fig:het}214\end{figure}215216\begin{table}[p]217\centering218\begin{threeparttable}219\caption{Heterogeneity of vertical inequity}220\label{tab:het}221\small222\input{../results/tables/heterogeneity}223\begin{tablenotes}[flushleft]\footnotesize224\item \textit{Notes:} Cheng fixed-effects estimates of225$\gamma = \beta - 1$ by subsample; cells with fewer than 20 sales after226masking are dropped. Municipality-clustered standard errors in227parentheses. *, **, *** denote significance at 10\%, 5\%, 1\%.228\end{tablenotes}229\end{threeparttable}230\end{table}231232\subsection{Horizontal inequity}233234Table~\ref{tab:horizontal} reports the dispersion regression. The mean235absolute deviation of a sale's log ratio from its cell median is 0.188 ---236the typical Quebec dwelling is assessed 19\% away from its market's norm,237an enormous horizontal lottery consistent with the COD evidence. The238deviation rises by 0.75~log points per decade of building age and by2395.2~points for cottages, and falls by 5.5~points for condominiums.240Old, idiosyncratic, land-heavy properties receive not only biased but241\emph{noisy} valuations.242243\begin{table}[t]244\centering245\begin{threeparttable}246\caption{Horizontal inequity: who receives noisy assessments?}247\label{tab:horizontal}248\small249\input{../results/tables/horizontal}250\begin{tablenotes}[flushleft]\footnotesize251\item \textit{Notes:} Dependent variable:252$|\ln r_i - \operatorname{med}_{c(i)} \ln r|$. Absorbing least squares253with cell fixed effects; municipality-clustered standard errors.254Omitted class: single-family. *, **, *** denote significance at 10\%,2555\%, 1\%.256\end{tablenotes}257\end{threeparttable}258\end{table}259260\subsection{The implied tax shift}261262Figure~\ref{fig:taxshift} and Table~\ref{tab:taxshift} translate the263estimates into tax dollars. Within the median market, the median dwelling264in the bottom price decile is assessed 65\% above the local norm --- and265therefore pays 65\% more property tax than uniform assessment would imply.266The overpayment falls to $+7\%$ in the second decile and crosses zero at267the median; the top decile underpays by about 5\%. The redistribution is268strongly convex: the burden of assessment error is overwhelmingly269concentrated on the cheapest tenth of the housing stock --- precisely the270segment where owners have the least capacity to appeal and the highest271housing-cost burdens. The magnitudes rival those documented for the U.S.272by \citet{berry2021reassessing}, in a system with none of the U.S.'s273institutional fragmentation.274275\begin{figure}[t]276\centering277\includegraphics[width=0.85\textwidth]{fig_taxshift.png}278\caption{The implied property-tax shift. Median relative assessment error279$e_i = r_i/\operatorname{med}_{c(i)}(r) - 1$ by within-market sale-price280decile. Because the levy is proportional to the rolled value, $e_i$ equals281the percentage over- or under-payment relative to uniform assessment.}282\label{fig:taxshift}283\end{figure}284285\begin{table}[t]286\centering287\begin{threeparttable}288\caption{Excess tax burden by within-market price decile}289\label{tab:taxshift}290\small291\input{../results/tables/taxshift}292\begin{tablenotes}[flushleft]\footnotesize293\item \textit{Notes:} Relative assessment error by within-cell sale-price294decile. The mean is sensitive to extreme low-price sales; the median is295the preferred summary. Standard errors of the mean in parentheses.296\end{tablenotes}297\end{threeparttable}298\end{table}299