% Author: Simon-Pierre Boucher — contact@spboucher.ai
% ============================================================================
\section{Robustness, Heterogeneity, and Validation}
\label{sec:robustness}
% ============================================================================

\subsection{Stability of the implicit prices}
\label{subsec:stability}

Table~\ref{tab:robustness} re-estimates the grand FSA-fixed-effects model on six
alternative samples and specifications. The size elasticity stays within the narrow band
$0.51$--$0.62$ and the bathroom premium within $0.10$--$0.12$ log points across every cut:
restricting to houses, restricting to condominiums, tightening the price trim to the
0.5/99.5 percentiles, dropping FSAs with fewer than fifty listings, and limiting the
sample to the three largest provinces. The condominium subsample exhibits both the highest
fit ($R^2=0.81$) and the largest size elasticity, consistent with condominium prices being
more tightly pinned down by floor area and location and less by idiosyncratic lot and
structure features. The overall picture is one of striking parameter stability: the
headline implicit prices are not artefacts of a particular sample definition.

\input{../results/tables/robustness}

\subsection{Implicit prices along the price distribution}
\label{subsec:quantile}

OLS recovers the implicit price at the conditional mean, but buyers at the bottom and top
of the market may value attributes differently \citep{zietz2008determinants}. We estimate
quantile hedonic regressions \citep{koenker1978regression} at the 10th through 90th
percentiles of price (Table~\ref{tab:quantile}, Figure~\ref{fig:quantile}). Two patterns
stand out. The living-area elasticity is roughly flat-to-rising, climbing from $0.56$ at
the bottom to $0.60$ at the top, indicating that floor space is valued slightly more in
expensive segments. The lot elasticity rises more steeply across the distribution,
consistent with land being a luxury component of value. The bathroom premium is stable
around $0.10$--$0.13$ throughout. The mean-based estimates in Table~\ref{tab:regression}
are thus representative, but they mask economically sensible distributional variation.

\input{../results/tables/quantile}

\begin{figure}[t]\centering
\includegraphics[width=\textwidth]{fig_quantile.png}
\caption{Implicit prices across the conditional price distribution. Quantile estimates
(with 95\% confidence intervals) of the living-area elasticity (left) and the
full-bathroom premium (right); the dashed line is the OLS estimate.}
\label{fig:quantile}
\end{figure}

\subsection{Residual spatial autocorrelation}
\label{subsec:spatial}

A central justification for the neighbourhood fixed effects is that they should absorb the
spatial dependence that pervades raw housing residuals
\citep{anselin1988spatial,dubin1988estimation,basu1998analysis}. We test
this directly by computing Moran's~$I$ \citep{moran1950notes} on the residuals, using
row-standardised $k$-nearest-neighbour spatial weights ($k=10$) on a random sample of
15{,}000 listings. The structure-only model leaves enormous spatial autocorrelation in its
residuals, $I=0.46$ ($z=145$, $p<0.01$): nearby dwellings are mispriced in the same
direction---the signature of omitted location. The grand model with FSA fixed effects cuts
this to $I=0.08$ ($z=23$), an 82\% reduction, confirming that the neighbourhood effects
absorb the overwhelming majority of the spatial signal. Figure~\ref{fig:moran} contrasts
the two Moran scatterplots. The small residual autocorrelation that remains is
within-neighbourhood and could be addressed by finer geographies or an explicit spatial
model \citep{lesage2009introduction}, but it is an order of magnitude smaller than the
dependence the fixed effects remove, vindicating the design over a parametric spatial-lag
alternative \citep{gibbons2012mostly}.

\begin{figure}[t]\centering
\includegraphics[width=\textwidth]{fig_moran.png}
\caption{Moran scatterplots of model residuals against their spatial lag ($k=10$ nearest
neighbours, 15{,}000-listing sample). Left: structure-only model. Right: grand model with
FSA fixed effects. The slope is Moran's~$I$; it collapses from 0.46 to 0.08.}
\label{fig:moran}
\end{figure}

\subsection{Do structural prices transfer across space?}
\label{subsec:lopo}

As a demanding test of external validity we perform leave-one-province-out
cross-validation: the structural model is estimated on all provinces but one and used to
predict the held-out province, allowing only a province-specific intercept (the price
\emph{level} is not identified out of region). Table~\ref{tab:lopo} reports the
within-province $R^2$. The structural implicit prices transfer well to most of the
country, with a mean held-out $R^2$ of $0.36$ and values above $0.40$ for the large
central and western markets; transfer is weaker for the small Atlantic samples, where
idiosyncratic stock and thin data dominate. That structural prices generalise across
provinces---even as price \emph{levels} differ by a factor of nine---reinforces the
paper's central decomposition: structure is broadly priced the same everywhere, and it is
location that varies.

\input{../results/tables/lopo}

\subsection{Heterogeneity across provinces}
\label{subsec:heterogeneity}

Estimating the within-FSA model province by province reveals economically meaningful
heterogeneity in the size elasticity (Figure~\ref{fig:heterogeneity}). The elasticity is
lowest in the high-price coastal markets---about $0.49$ in British Columbia and $0.50$ in
Ontario---and highest in the Prairies, reaching $0.65$--$0.66$ in Saskatchewan and
Manitoba. The pattern is intuitive: where land and location dominate value (Vancouver,
Toronto), an extra square metre of structure adds proportionally less, whereas in
lower-priced markets the building itself is a larger share of value and floor space
carries more weight. The bathroom premium shows the mirror pattern, larger in the Prairies
and Atlantic provinces than in the coastal metros.

\begin{figure}[t]\centering
\includegraphics[width=0.8\textwidth]{fig_heterogeneity.png}
\caption{Living-area elasticity of price by province, estimated within FSAs, with 95\%
cluster-robust confidence intervals. The elasticity is smallest in the expensive coastal
markets and largest in the Prairies.}
\label{fig:heterogeneity}
\end{figure}

\subsection{Out-of-sample valuation accuracy}
\label{subsec:oos}

A hedonic model that fits in-sample need not predict well. Table~\ref{tab:oos} and
Figure~\ref{fig:oos} report performance on a randomly held-out 20\% of listings. The model
attains an out-of-sample $R^2$ of \textbf{0.764} on log price---essentially identical to
its in-sample fit, indicating negligible over-fitting despite the thousand-plus location
effects. In price levels, after Duan smearing, the \textbf{median absolute valuation error
is 15.8\%}, the mean is 22.5\%, and \textbf{59\% of held-out dwellings are priced within
$\pm$20\%} of their actual list price (34\% within $\pm$10\%). These figures are within
the accuracy bands reported in the mass-appraisal and automated-valuation literature
\citep{mccluskey2013prediction,clapp2003semiparametric} and establish the transparent
hedonic specification as a credible valuation benchmark \citep{mullainathan2017machine}.

\input{../results/tables/oos}

\begin{figure}[t]\centering
\includegraphics[width=0.7\textwidth]{fig_oos.png}
\caption{Out-of-sample valuation accuracy on the held-out test fold: the share of listings
priced within $\pm$10\%, between 10 and 20\%, and beyond 20\% of the actual list price.
The out-of-sample $R^2$ is reported in the title.}
\label{fig:oos}
\end{figure}

\subsection{Model fit and residual behaviour}
\label{subsec:fit}

Panel~(a) of Figure~\ref{fig:fit} plots predicted against actual log prices for the grand
model; the cloud hugs the 45-degree line. The residuals (panel~(b)) are approximately
Gaussian and centred on zero, with only mild heavy tails---typical of housing data and
accommodated by the cluster-robust inference. We interpret the remaining dispersion as a
combination of genuine idiosyncratic pricing, listing strategy, and dwelling-level quality
(age, renovations, finish) that the data do not record.

\begin{figure}[t]\centering
\begin{subfigure}{0.46\textwidth}\includegraphics[width=\textwidth]{fig_fit.png}
\caption{Predicted vs.\ actual}\end{subfigure}\hfill
\begin{subfigure}{0.52\textwidth}\includegraphics[width=\textwidth]{fig_resid.png}
\caption{Residual diagnostics}\end{subfigure}
\caption{Grand-model (M5) goodness of fit and residual behaviour.}
\label{fig:fit}
\end{figure}
