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

Table~\ref{tab:robust} and Figure~\ref{fig:robust} re-estimate the two
headline elasticities --- $\gamma_{\text{FE}}$ and the Clapp
$\gamma_{\text{IV}}$ --- across ten sample and measurement variants;
Table~\ref{tab:quantile} reports the quantile coefficients underlying
Figure~\ref{fig:quantile}.

\begin{table}[t]
\centering
\begin{threeparttable}
\caption{Robustness of the vertical-inequity elasticities}
\label{tab:robust}
\small
\input{../results/tables/robustness}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes:} Each row re-estimates the Cheng fixed-effects and
Clapp IV regressions on the indicated variant; cells with fewer than 20
sales after restriction are dropped. Municipality-clustered standard
errors in parentheses. *, **, *** denote significance at 10\%, 5\%, 1\%
against $\gamma = 0$.
\end{tablenotes}
\end{threeparttable}
\end{table}

\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{fig_robustness.png}
\caption{Stability of $\gamma$ across sample variants. Filled circles:
fixed-effects estimates; open squares: Clapp IV. Whiskers are 95\%
confidence intervals, municipality-clustered.}
\label{fig:robust}
\end{figure}

\begin{table}[t]
\centering
\begin{threeparttable}
\caption{Quantile-regression coefficients}
\label{tab:quantile}
\small
\input{../results/tables/quantile}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes:} Quantile regressions of within-cell demeaned
$\ln AV$ on demeaned $\ln SP$, estimated on a seeded 250{,}000-sale
subsample. Stars test $H_0\!:\beta(\tau)=1$.
\end{tablenotes}
\end{threeparttable}
\end{table}

\paragraph{Match quality.} Requiring the maximum matcher score of 220
(210{,}820 sales) or a match distance under 10~metres changes
$\gamma_{\text{FE}}$ to $-0.41$ and $-0.36$ respectively, and
$\gamma_{\text{IV}}$ to $-0.10$ and $-0.09$: if anything, the cleanest
matches show \emph{more} regressivity, ruling out mismatch noise as the
source.

\paragraph{Extreme sales.} Dropping all sales under \$100{,}000 --- the
segment most likely to harbour residual non-arm's-length transfers ---
attenuates the FE estimate modestly ($-0.30$) and the IV to $-0.05$, both
still overwhelmingly significant. Tightening the ratio trim to the 5th--95th
percentiles, which mechanically compresses dispersion, cuts
$\gamma_{\text{FE}}$ to $-0.15$ and $\gamma_{\text{IV}}$ to $-0.04$; the
qualitative conclusion is unchanged, and the ordering
$|\gamma_{\text{IV}}| < |\gamma_{\text{FE}}|$ is preserved in every
variant, as the measurement-error logic requires.

\paragraph{Composition.} Single-family homes alone give
$\gamma_{\text{FE}} = -0.45$; condominiums alone $-0.12$ (with an IV
estimate indistinguishable from zero), confirming that the aggregate
result is not an artefact of pooling heterogeneous property classes ---
each class is regressive or neutral on its own, none is progressive.

\paragraph{Time and market depth.} Splitting the window into 2021--2023
and 2024--2026 yields $-0.35$ and $-0.34$ (FE); restricting to
municipalities with at least 300 sales, or to cells with at least 50,
moves the estimates by less than 0.005. The phenomenon is stable across
the largest housing-cycle swing in recent Canadian history.

\paragraph{Inference.} With 625 municipality clusters, the
cluster-robust $t$-statistics on $\gamma_{\text{FE}}$ exceed 11 in every
variant; the IV first stage is enormous (the rank instrument correlates
with within-cell log price at $F$ far above conventional thresholds), so
weak-instrument concerns do not arise.