spb/wp9_uqo Public

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Results}4\label{sec:results}5% ============================================================================67\subsection{The value of location}8\label{subsec:location}910Figure~\ref{fig:r2} summarises the explanatory power of the specification ladder. The11structural-only model (M1) accounts for 46.4\% of the variation in log prices. Adding12dwelling-type and ownership controls (M2) barely moves the fit, but introducing province13fixed effects (M3) raises $R^2$ to 56.8\%, and resolving location at the FSA scale lifts14it to \textbf{76.2\%} for houses (M4) and \textbf{76.7\%} for the grand model (M5). The15implication is stark: moving from province to neighbourhood resolution adds about twenty16percentage points of explained variance, and \emph{location as a whole accounts for17roughly thirty percentage points}---more than the entire structural bundle. This is the18central result of the paper and a quantitative statement of the realtor's adage that what19matters is ``location, location, location.''2021\begin{figure}[t]\centering22\includegraphics[width=0.82\textwidth]{fig_r2.png}23\caption{Share of log-price variation explained ($R^2$) across the five nested24specifications. The jump from M3 (province) to M4/M5 (FSA) quantifies the value of25resolving location at the neighbourhood scale.}26\label{fig:r2}27\end{figure}2829\subsection{Implicit prices of structural attributes}30\label{subsec:implicit}3132Table~\ref{tab:regression} reports the regression estimates. The living-area elasticity is33remarkably stable across specifications, at $0.66$ in the raw structural model and34settling near $0.55$ once location is controlled for: a 10\% larger dwelling commands35roughly a 5.5\% higher price. The slight decline as controls are added is consistent with36larger homes being located in more expensive areas---a confound the FSA effects remove.3738Full bathrooms carry one of the strongest structural premia: about $0.11$ log points in39the grand model, or roughly an \textbf{11\% price increase} per additional bathroom,40rising to nearly 15\% in the province-FE model. Half bathrooms attract a small41\emph{negative} conditional coefficient, which we read not as a disamenity but as a proxy42for older or more compartmentalised floor plans once total area and full baths are held43fixed. Bedrooms are economically negligible conditional on living area: holding floor44space constant, subdividing it into more bedrooms does not raise value---a textbook45hedonic finding \citep{sirmans2005composition} that recurs across our samples. Lot information enters positively ($\ln$46lot elasticity around $0.03$) but modestly, reflecting both the noisiness of the parsed47lot field and the fact that, within a neighbourhood, lot variation is compressed.4849Figure~\ref{fig:forest} presents the structural implicit prices with 95\% cluster-robust50confidence intervals, making visually plain the dominance of living area and bathrooms and51the near-zero conditional effects of bedrooms, parking and storeys.5253\input{../results/tables/regression}5455\begin{figure}[t]\centering56\includegraphics[width=0.82\textwidth]{fig_forest.png}57\caption{Marginal implicit prices of structural attributes (houses, province-FE model M3)58with 95\% cluster-robust confidence intervals, on the log-price scale.}59\label{fig:forest}60\end{figure}6162The price--size gradient by dwelling type (Figure~\ref{fig:size_gradient}) confirms the63log--log structure: median price rises concavely with living area for both dwelling types.64Unconditionally, condominiums list \emph{above} houses of the same size---a compositional65effect of their concentration in the expensive metropolitan markets---whereas conditional66on location and ownership the estimated dwelling-type effects show that houses command the67premium, in line with Table~\ref{tab:regression}.6869\begin{figure}[t]\centering70\includegraphics[width=0.78\textwidth]{fig_size_gradient.png}71\caption{Median price by living-area bin and dwelling type. The concave gradient is72linearised by the semi-log specification.}73\label{fig:size_gradient}74\end{figure}7576\subsection{The geography of housing value}77\label{subsec:geography}7879Figure~\ref{fig:maps} maps the spatial structure of value directly. The familiar outline80of populated Canada emerges from the listing coordinates. The Vancouver corridor and the81Greater Toronto--Golden Horseshoe area sit at the top of the price-per-square-metre82distribution, while the Prairies and Atlantic Canada anchor the bottom. The right panel83aggregates to FSA medians, the geographic unit absorbed in the grand model.8485\begin{figure}[t]\centering86\begin{subfigure}{0.49\textwidth}\includegraphics[width=\textwidth]{fig_map.png}87\caption{All listings}\end{subfigure}\hfill88\begin{subfigure}{0.49\textwidth}\includegraphics[width=\textwidth]{fig_fsa_map.png}89\caption{FSA neighbourhood medians}\end{subfigure}90\caption{Spatial distribution of housing value. Colour encodes log price per m$^2$; bubble91area in panel~(b) is proportional to $\sqrt{\text{listings}}$. Labels mark the provinces92with substantial samples; the data cover nine provinces, while the territories and Prince93Edward Island contain no listings.}94\label{fig:maps}95\end{figure}9697To translate location into a clean dollar statement we recover each FSA's fixed effect98from the grand model---its price premium net of structure, dwelling type and99ownership---and express it relative to the national median (Figure~\ref{fig:premia}). The100highest-valued neighbourhoods, all in the City of Vancouver, trade at \textbf{150--200\%101above} the national-median neighbourhood for an otherwise identical dwelling; the lowest,102in rural Saskatchewan, Manitoba and Newfoundland, sit \textbf{60--67\% below}. The full103premium distribution thus spans a factor of roughly nine between the most and least104expensive neighbourhoods, dwarfing the price range attributable to any single structural105attribute.106107\begin{figure}[t]\centering108\includegraphics[width=0.78\textwidth]{fig_premia.png}109\caption{Highest- and lowest-valued neighbourhoods (FSAs) in Canada, measured as the110estimated location premium relative to the national-median neighbourhood, net of111structure, dwelling type and ownership. Only FSAs with at least 50 listings are shown.}112\label{fig:premia}113\end{figure}114115\subsection{Decomposing the variance of prices}116\label{subsec:decomp}117118Figure~\ref{fig:decomp} casts the specification ladder as a decomposition of the variance119of log prices into the share explained by each successive block of controls. Physical120structure accounts for 46\% of the variance; dwelling type and ownership add a further1210.4~points; province adds about 10~points; and resolving location to the neighbourhood122adds a further 20~points, for a total location contribution near 30~points. Just under a123quarter of the variance remains unexplained and is attributable to idiosyncratic pricing124and unobserved dwelling quality. The visual makes the headline unmistakable: the single125largest identified block of housing value in Canada is the neighbourhood.126127\begin{figure}[t]\centering128\includegraphics[width=0.92\textwidth]{fig_decomp.png}129\caption{Variance decomposition of Canadian log house prices into structure, dwelling130type/ownership, province, neighbourhood (FSA) and the unexplained residual, from the131nested specification ladder.}132\label{fig:decomp}133\end{figure}134135\subsection{Nonlinearity: diminishing returns to floor space}136\label{subsec:nonlinear}137138The constant-elasticity assumption is convenient but restrictive. Re-estimating the grand139model with a quadratic in log living area yields a positive linear term and a140significantly negative quadratic term ($\widehat{\beta}_1=1.06$,141$\widehat{\beta}_2=-0.053$), implying that the marginal elasticity of price with respect142to floor space \emph{declines} with dwelling size. Figure~\ref{fig:nonlinear} traces the143implied marginal elasticity: it falls from roughly $0.65$ for a compact 60~m$^2$ dwelling144to about $0.45$ for a large 350~m$^2$ home. Economically, the first square metres of145living space are valued most highly and additional space is subject to diminishing146returns---consistent with the nonparametric hedonic surfaces of \citet{mcmillen2010issues}.147The constant-elasticity estimate of $0.55$ is best read as an average over the size148distribution.149150\begin{figure}[t]\centering151\includegraphics[width=0.78\textwidth]{fig_nonlinear.png}152\caption{Marginal elasticity of price with respect to living area as a function of153dwelling size, from a grand model with a quadratic in log area (95\% cluster-robust band).154The dashed line is the constant-elasticity estimate.}155\label{fig:nonlinear}156\end{figure}157158\subsection{The urban price gradient}159\label{subsec:gradient}160161Classic urban theory predicts that, holding structure fixed, value declines with distance162from employment centres163\citep{alonso1964location,muth1969cities,mills1967aggregative,ahlfeldt2015density}. We164compute the great-circle distance from each listing to the165nearest of nine major Canadian metropolitan centres and relate it to the location166component of price (the residual from a structure-only model). Figure~\ref{fig:gradient}167confirms a pronounced gradient: dwellings within the metropolitan core carry location168premia of tens of percent over the structure-only benchmark, and the premium decays169steadily with distance, turning negative beyond roughly 80~km. A log-linear fit implies a170semi-elasticity of $-0.085$ ($t>100$): a doubling of distance to the nearest metro is171associated with an 8.5\% lower location premium. This agglomeration gradient is precisely172the force that the FSA fixed effects absorb non-parametrically in the grand model173\citep{combes2015empirics}.174175\begin{figure}[t]\centering176\includegraphics[width=0.78\textwidth]{fig_gradient.png}177\caption{Urban price gradient: the location component of price (relative to a178structure-only benchmark) against distance to the nearest of nine major Canadian metros.179The horizontal axis is on a symmetric-log scale.}180\label{fig:gradient}181\end{figure}182