spb/wp9_uqo Public

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Robustness, Heterogeneity, and Validation}4\label{sec:robustness}5% ============================================================================67\subsection{Stability of the implicit prices}8\label{subsec:stability}910Table~\ref{tab:robustness} re-estimates the grand FSA-fixed-effects model on six11alternative samples and specifications. The size elasticity stays within the narrow band12$0.51$--$0.62$ and the bathroom premium within $0.10$--$0.12$ log points across every cut:13restricting to houses, restricting to condominiums, tightening the price trim to the140.5/99.5 percentiles, dropping FSAs with fewer than fifty listings, and limiting the15sample to the three largest provinces. The condominium subsample exhibits both the highest16fit ($R^2=0.81$) and the largest size elasticity, consistent with condominium prices being17more tightly pinned down by floor area and location and less by idiosyncratic lot and18structure features. The overall picture is one of striking parameter stability: the19headline implicit prices are not artefacts of a particular sample definition.2021\input{../results/tables/robustness}2223\subsection{Implicit prices along the price distribution}24\label{subsec:quantile}2526OLS recovers the implicit price at the conditional mean, but buyers at the bottom and top27of the market may value attributes differently \citep{zietz2008determinants}. We estimate28quantile hedonic regressions \citep{koenker1978regression} at the 10th through 90th29percentiles of price (Table~\ref{tab:quantile}, Figure~\ref{fig:quantile}). Two patterns30stand out. The living-area elasticity is roughly flat-to-rising, climbing from $0.56$ at31the bottom to $0.60$ at the top, indicating that floor space is valued slightly more in32expensive segments. The lot elasticity rises more steeply across the distribution,33consistent with land being a luxury component of value. The bathroom premium is stable34around $0.10$--$0.13$ throughout. The mean-based estimates in Table~\ref{tab:regression}35are thus representative, but they mask economically sensible distributional variation.3637\input{../results/tables/quantile}3839\begin{figure}[t]\centering40\includegraphics[width=\textwidth]{fig_quantile.png}41\caption{Implicit prices across the conditional price distribution. Quantile estimates42(with 95\% confidence intervals) of the living-area elasticity (left) and the43full-bathroom premium (right); the dashed line is the OLS estimate.}44\label{fig:quantile}45\end{figure}4647\subsection{Residual spatial autocorrelation}48\label{subsec:spatial}4950A central justification for the neighbourhood fixed effects is that they should absorb the51spatial dependence that pervades raw housing residuals52\citep{anselin1988spatial,dubin1988estimation,basu1998analysis}. We test53this directly by computing Moran's~$I$ \citep{moran1950notes} on the residuals, using54row-standardised $k$-nearest-neighbour spatial weights ($k=10$) on a random sample of5515{,}000 listings. The structure-only model leaves enormous spatial autocorrelation in its56residuals, $I=0.46$ ($z=145$, $p<0.01$): nearby dwellings are mispriced in the same57direction---the signature of omitted location. The grand model with FSA fixed effects cuts58this to $I=0.08$ ($z=23$), an 82\% reduction, confirming that the neighbourhood effects59absorb the overwhelming majority of the spatial signal. Figure~\ref{fig:moran} contrasts60the two Moran scatterplots. The small residual autocorrelation that remains is61within-neighbourhood and could be addressed by finer geographies or an explicit spatial62model \citep{lesage2009introduction}, but it is an order of magnitude smaller than the63dependence the fixed effects remove, vindicating the design over a parametric spatial-lag64alternative \citep{gibbons2012mostly}.6566\begin{figure}[t]\centering67\includegraphics[width=\textwidth]{fig_moran.png}68\caption{Moran scatterplots of model residuals against their spatial lag ($k=10$ nearest69neighbours, 15{,}000-listing sample). Left: structure-only model. Right: grand model with70FSA fixed effects. The slope is Moran's~$I$; it collapses from 0.46 to 0.08.}71\label{fig:moran}72\end{figure}7374\subsection{Do structural prices transfer across space?}75\label{subsec:lopo}7677As a demanding test of external validity we perform leave-one-province-out78cross-validation: the structural model is estimated on all provinces but one and used to79predict the held-out province, allowing only a province-specific intercept (the price80\emph{level} is not identified out of region). Table~\ref{tab:lopo} reports the81within-province $R^2$. The structural implicit prices transfer well to most of the82country, with a mean held-out $R^2$ of $0.36$ and values above $0.40$ for the large83central and western markets; transfer is weaker for the small Atlantic samples, where84idiosyncratic stock and thin data dominate. That structural prices generalise across85provinces---even as price \emph{levels} differ by a factor of nine---reinforces the86paper's central decomposition: structure is broadly priced the same everywhere, and it is87location that varies.8889\input{../results/tables/lopo}9091\subsection{Heterogeneity across provinces}92\label{subsec:heterogeneity}9394Estimating the within-FSA model province by province reveals economically meaningful95heterogeneity in the size elasticity (Figure~\ref{fig:heterogeneity}). The elasticity is96lowest in the high-price coastal markets---about $0.49$ in British Columbia and $0.50$ in97Ontario---and highest in the Prairies, reaching $0.65$--$0.66$ in Saskatchewan and98Manitoba. The pattern is intuitive: where land and location dominate value (Vancouver,99Toronto), an extra square metre of structure adds proportionally less, whereas in100lower-priced markets the building itself is a larger share of value and floor space101carries more weight. The bathroom premium shows the mirror pattern, larger in the Prairies102and Atlantic provinces than in the coastal metros.103104\begin{figure}[t]\centering105\includegraphics[width=0.8\textwidth]{fig_heterogeneity.png}106\caption{Living-area elasticity of price by province, estimated within FSAs, with 95\%107cluster-robust confidence intervals. The elasticity is smallest in the expensive coastal108markets and largest in the Prairies.}109\label{fig:heterogeneity}110\end{figure}111112\subsection{Out-of-sample valuation accuracy}113\label{subsec:oos}114115A hedonic model that fits in-sample need not predict well. Table~\ref{tab:oos} and116Figure~\ref{fig:oos} report performance on a randomly held-out 20\% of listings. The model117attains an out-of-sample $R^2$ of \textbf{0.764} on log price---essentially identical to118its in-sample fit, indicating negligible over-fitting despite the thousand-plus location119effects. In price levels, after Duan smearing, the \textbf{median absolute valuation error120is 15.8\%}, the mean is 22.5\%, and \textbf{59\% of held-out dwellings are priced within121$\pm$20\%} of their actual list price (34\% within $\pm$10\%). These figures are within122the accuracy bands reported in the mass-appraisal and automated-valuation literature123\citep{mccluskey2013prediction,clapp2003semiparametric} and establish the transparent124hedonic specification as a credible valuation benchmark \citep{mullainathan2017machine}.125126\input{../results/tables/oos}127128\begin{figure}[t]\centering129\includegraphics[width=0.7\textwidth]{fig_oos.png}130\caption{Out-of-sample valuation accuracy on the held-out test fold: the share of listings131priced within $\pm$10\%, between 10 and 20\%, and beyond 20\% of the actual list price.132The out-of-sample $R^2$ is reported in the title.}133\label{fig:oos}134\end{figure}135136\subsection{Model fit and residual behaviour}137\label{subsec:fit}138139Panel~(a) of Figure~\ref{fig:fit} plots predicted against actual log prices for the grand140model; the cloud hugs the 45-degree line. The residuals (panel~(b)) are approximately141Gaussian and centred on zero, with only mild heavy tails---typical of housing data and142accommodated by the cluster-robust inference. We interpret the remaining dispersion as a143combination of genuine idiosyncratic pricing, listing strategy, and dwelling-level quality144(age, renovations, finish) that the data do not record.145146\begin{figure}[t]\centering147\begin{subfigure}{0.46\textwidth}\includegraphics[width=\textwidth]{fig_fit.png}148\caption{Predicted vs.\ actual}\end{subfigure}\hfill149\begin{subfigure}{0.52\textwidth}\includegraphics[width=\textwidth]{fig_resid.png}150\caption{Residual diagnostics}\end{subfigure}151\caption{Grand-model (M5) goodness of fit and residual behaviour.}152\label{fig:fit}153\end{figure}154