spb/wp10_uqo Public

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Methodology}4\label{sec:method}56Let $AV_i$ denote the assessed value of property $i$ on the roll in force7at its sale, $SP_i$ its sale price, and $r_i = AV_i/SP_i$ the assessment8ratio. Under perfectly proportional (vertically equitable) assessment,9$\mathbb{E}[r_i \mid SP_i]$ is constant. All tests below ask, in different10ways, whether $r_i$ instead declines with value.1112\subsection{IAAO ratio-study diagnostics}1314For any group of sales we report the four statistics of the IAAO15\textit{Standard on Ratio Studies} \citep{iaao2013standard}. The16\emph{median ratio} measures the assessment level. The \emph{coefficient of17dispersion},18$\mathrm{COD} = 100 \cdot \operatorname{mean}\!\left(|r_i -19\tilde r|\right) / \tilde r$ with $\tilde r$ the median ratio, measures20horizontal uniformity; the standard deems residential CODs above 1521unacceptable. The \emph{price-related differential},22$\mathrm{PRD} = \bar r \,/\, (\sum_i AV_i / \sum_i SP_i)$, compares the23unweighted and value-weighted mean ratios; PRD~$>1.03$ indicates24regressivity. Because the PRD is sensitive to outliers, the standard's25preferred vertical measure is the \emph{coefficient of price-related bias}26(PRB), the slope $b$ in27\begin{equation}28\frac{r_i - \tilde r}{\tilde r}29  \;=\; a + b \cdot \log_2\!\Big(\tfrac{1}{2}SP_i +30  \tfrac{1}{2}\,AV_i/\tilde r\Big) + u_i ,31\label{eq:prb}32\end{equation}33which measures the proportional change in the ratio per doubling of value;34the acceptable band is $[-0.05, 0.05]$. We compute percentile-bootstrap35confidence intervals for all four statistics. Because ratio levels drift36with roll staleness, municipality-level diagnostics are computed within37municipality~$\times$~sale-year blocks --- inside which a single roll is in38force --- and aggregated across years by the median.3940\subsection{Regression tests with market-timing fixed effects}4142Our workhorse is the log-log specification of \citet{cheng1974property},43\begin{equation}44\ln AV_i \;=\; \alpha_{c(i)} + \beta \, \ln SP_i + \varepsilon_i ,45\label{eq:cheng}46\end{equation}47where $c(i)$ indexes the municipality~$\times$~roll~$\times$~sale-year cell48of sale $i$ and the $\alpha_c$ are 2{,}884 absorbed fixed effects.49Proportionality implies $\beta = 1$; we report50$\gamma \equiv \beta - 1$, the elasticity of the assessment \emph{ratio}51with respect to price ($\gamma<0$: regressive). The fixed effects guarantee52that $\gamma$ is identified only from comparisons of dwellings assessed by53the same authority, on the same roll vintage, and sold in the same year ---54purging the staleness drift, municipal composition, and aggregate market55movements in one stroke. For completeness we also report the pooled56regression without fixed effects and the levels test of57\citet{paglin1972equity}. Inference is clustered at the municipality level58throughout (625 clusters).5960\subsection{Measurement error and the Clapp instrument}6162Sale prices measure market value with idiosyncratic noise --- bilateral63bargaining, unobserved conditions of sale --- so OLS on64equation~\eqref{eq:cheng} suffers attenuation bias: $\hat\beta < 1$ even65under proportional assessment \citep{kochin1982vertical, kennedy1984unfair}.66Following \citet{clapp1990new}, we instrument $\ln SP_i$ with a coarse rank67variable $Z_i \in \{-1, 0, +1\}$ that flags sales in the bottom or top68third of \emph{both} the within-cell $\ln AV$ and $\ln SP$ distributions.69Because $Z_i$ retains only ordinal information agreed on by both measures of70value, it is (nearly) orthogonal to the transitory component of either, and71the resulting two-stage least-squares estimate of $\beta$ is consistent72under classical measurement error. The demeaned-within-cell implementation73preserves the fixed-effects structure. Throughout the paper we treat74$\gamma_{\text{IV}}$ as a conservative lower bound on regressivity and75$\gamma_{\text{FE}}$ as the descriptive upper bound; the truth lies between.7677\subsection{Quantile profile and heterogeneity}7879Averaging can hide where proportionality fails. We estimate80equation~\eqref{eq:cheng} by quantile regression on within-cell demeaned81variables at $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$82\citep{mcmillen2020assessment}, tracing $\beta(\tau)$ across the83conditional distribution of assessed values. We then re-estimate the84fixed-effects model on subsamples --- property class, building-age bands,85assessed-land-share bands, roll-lag bands, municipality size, and sale year86--- keeping only cells that retain at least 20 sales, to locate the87inequity where the mass-appraisal problem is hardest.8889\subsection{Horizontal inequity}9091Vertical tests concern the \emph{mean} of $r_i$ given value; horizontal92equity concerns its \emph{dispersion} among comparable properties93\citep{allen2002measuring, sirmans2008vertical}. Beyond the COD, we ask94who receives noisy assessments: we regress the absolute deviation of a95sale's log ratio from its cell median,96$|\ln r_i - \operatorname{med}_{c(i)} \ln r|$, on property characteristics97(age, land share, property-class indicators) with cell fixed effects.9899\subsection{The implied tax shift}100101Because Quebec taxes the rolled value without exemptions or caps102(Section~\ref{sec:inst}), a property whose ratio exceeds its jurisdiction's103median by $x\%$ pays exactly $x\%$ more tax than uniform assessment would104imply. For each sale we compute the relative assessment error105$e_i = r_i / \operatorname{med}_{c(i)}(r) - 1$ and average it by106within-cell sale-price decile. This translates the econometrics into the107policy-relevant object: the percentage over- or under-payment of property108tax by position in the local price distribution.109