% Author: Simon-Pierre Boucher — contact@spboucher.ai
% ============================================================================
\section{Methodology}
\label{sec:method}

Let $AV_i$ denote the assessed value of property $i$ on the roll in force
at its sale, $SP_i$ its sale price, and $r_i = AV_i/SP_i$ the assessment
ratio. Under perfectly proportional (vertically equitable) assessment,
$\mathbb{E}[r_i \mid SP_i]$ is constant. All tests below ask, in different
ways, whether $r_i$ instead declines with value.

\subsection{IAAO ratio-study diagnostics}

For any group of sales we report the four statistics of the IAAO
\textit{Standard on Ratio Studies} \citep{iaao2013standard}. The
\emph{median ratio} measures the assessment level. The \emph{coefficient of
dispersion},
$\mathrm{COD} = 100 \cdot \operatorname{mean}\!\left(|r_i -
\tilde r|\right) / \tilde r$ with $\tilde r$ the median ratio, measures
horizontal uniformity; the standard deems residential CODs above 15
unacceptable. The \emph{price-related differential},
$\mathrm{PRD} = \bar r \,/\, (\sum_i AV_i / \sum_i SP_i)$, compares the
unweighted and value-weighted mean ratios; PRD~$>1.03$ indicates
regressivity. Because the PRD is sensitive to outliers, the standard's
preferred vertical measure is the \emph{coefficient of price-related bias}
(PRB), the slope $b$ in
\begin{equation}
\frac{r_i - \tilde r}{\tilde r}
  \;=\; a + b \cdot \log_2\!\Big(\tfrac{1}{2}SP_i +
  \tfrac{1}{2}\,AV_i/\tilde r\Big) + u_i ,
\label{eq:prb}
\end{equation}
which measures the proportional change in the ratio per doubling of value;
the acceptable band is $[-0.05, 0.05]$. We compute percentile-bootstrap
confidence intervals for all four statistics. Because ratio levels drift
with roll staleness, municipality-level diagnostics are computed within
municipality~$\times$~sale-year blocks --- inside which a single roll is in
force --- and aggregated across years by the median.

\subsection{Regression tests with market-timing fixed effects}

Our workhorse is the log-log specification of \citet{cheng1974property},
\begin{equation}
\ln AV_i \;=\; \alpha_{c(i)} + \beta \, \ln SP_i + \varepsilon_i ,
\label{eq:cheng}
\end{equation}
where $c(i)$ indexes the municipality~$\times$~roll~$\times$~sale-year cell
of sale $i$ and the $\alpha_c$ are 2{,}884 absorbed fixed effects.
Proportionality implies $\beta = 1$; we report
$\gamma \equiv \beta - 1$, the elasticity of the assessment \emph{ratio}
with respect to price ($\gamma<0$: regressive). The fixed effects guarantee
that $\gamma$ is identified only from comparisons of dwellings assessed by
the same authority, on the same roll vintage, and sold in the same year ---
purging the staleness drift, municipal composition, and aggregate market
movements in one stroke. For completeness we also report the pooled
regression without fixed effects and the levels test of
\citet{paglin1972equity}. Inference is clustered at the municipality level
throughout (625 clusters).

\subsection{Measurement error and the Clapp instrument}

Sale prices measure market value with idiosyncratic noise --- bilateral
bargaining, unobserved conditions of sale --- so OLS on
equation~\eqref{eq:cheng} suffers attenuation bias: $\hat\beta < 1$ even
under proportional assessment \citep{kochin1982vertical, kennedy1984unfair}.
Following \citet{clapp1990new}, we instrument $\ln SP_i$ with a coarse rank
variable $Z_i \in \{-1, 0, +1\}$ that flags sales in the bottom or top
third of \emph{both} the within-cell $\ln AV$ and $\ln SP$ distributions.
Because $Z_i$ retains only ordinal information agreed on by both measures of
value, it is (nearly) orthogonal to the transitory component of either, and
the resulting two-stage least-squares estimate of $\beta$ is consistent
under classical measurement error. The demeaned-within-cell implementation
preserves the fixed-effects structure. Throughout the paper we treat
$\gamma_{\text{IV}}$ as a conservative lower bound on regressivity and
$\gamma_{\text{FE}}$ as the descriptive upper bound; the truth lies between.

\subsection{Quantile profile and heterogeneity}

Averaging can hide where proportionality fails. We estimate
equation~\eqref{eq:cheng} by quantile regression on within-cell demeaned
variables at $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$
\citep{mcmillen2020assessment}, tracing $\beta(\tau)$ across the
conditional distribution of assessed values. We then re-estimate the
fixed-effects model on subsamples --- property class, building-age bands,
assessed-land-share bands, roll-lag bands, municipality size, and sale year
--- keeping only cells that retain at least 20 sales, to locate the
inequity where the mass-appraisal problem is hardest.

\subsection{Horizontal inequity}

Vertical tests concern the \emph{mean} of $r_i$ given value; horizontal
equity concerns its \emph{dispersion} among comparable properties
\citep{allen2002measuring, sirmans2008vertical}. Beyond the COD, we ask
who receives noisy assessments: we regress the absolute deviation of a
sale's log ratio from its cell median,
$|\ln r_i - \operatorname{med}_{c(i)} \ln r|$, on property characteristics
(age, land share, property-class indicators) with cell fixed effects.

\subsection{The implied tax shift}

Because Quebec taxes the rolled value without exemptions or caps
(Section~\ref{sec:inst}), a property whose ratio exceeds its jurisdiction's
median by $x\%$ pays exactly $x\%$ more tax than uniform assessment would
imply. For each sale we compute the relative assessment error
$e_i = r_i / \operatorname{med}_{c(i)}(r) - 1$ and average it by
within-cell sale-price decile. This translates the econometrics into the
policy-relevant object: the percentage over- or under-payment of property
tax by position in the local price distribution.
