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

\subsection{The value of location}
\label{subsec:location}

Figure~\ref{fig:r2} summarises the explanatory power of the specification ladder. The
structural-only model (M1) accounts for 46.4\% of the variation in log prices. Adding
dwelling-type and ownership controls (M2) barely moves the fit, but introducing province
fixed effects (M3) raises $R^2$ to 56.8\%, and resolving location at the FSA scale lifts
it to \textbf{76.2\%} for houses (M4) and \textbf{76.7\%} for the grand model (M5). The
implication is stark: moving from province to neighbourhood resolution adds about twenty
percentage points of explained variance, and \emph{location as a whole accounts for
roughly thirty percentage points}---more than the entire structural bundle. This is the
central result of the paper and a quantitative statement of the realtor's adage that what
matters is ``location, location, location.''

\begin{figure}[t]\centering
\includegraphics[width=0.82\textwidth]{fig_r2.png}
\caption{Share of log-price variation explained ($R^2$) across the five nested
specifications. The jump from M3 (province) to M4/M5 (FSA) quantifies the value of
resolving location at the neighbourhood scale.}
\label{fig:r2}
\end{figure}

\subsection{Implicit prices of structural attributes}
\label{subsec:implicit}

Table~\ref{tab:regression} reports the regression estimates. The living-area elasticity is
remarkably stable across specifications, at $0.66$ in the raw structural model and
settling near $0.55$ once location is controlled for: a 10\% larger dwelling commands
roughly a 5.5\% higher price. The slight decline as controls are added is consistent with
larger homes being located in more expensive areas---a confound the FSA effects remove.

Full bathrooms carry one of the strongest structural premia: about $0.11$ log points in
the grand model, or roughly an \textbf{11\% price increase} per additional bathroom,
rising to nearly 15\% in the province-FE model. Half bathrooms attract a small
\emph{negative} conditional coefficient, which we read not as a disamenity but as a proxy
for older or more compartmentalised floor plans once total area and full baths are held
fixed. Bedrooms are economically negligible conditional on living area: holding floor
space constant, subdividing it into more bedrooms does not raise value---a textbook
hedonic finding \citep{sirmans2005composition} that recurs across our samples. Lot information enters positively ($\ln$
lot elasticity around $0.03$) but modestly, reflecting both the noisiness of the parsed
lot field and the fact that, within a neighbourhood, lot variation is compressed.

Figure~\ref{fig:forest} presents the structural implicit prices with 95\% cluster-robust
confidence intervals, making visually plain the dominance of living area and bathrooms and
the near-zero conditional effects of bedrooms, parking and storeys.

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

\begin{figure}[t]\centering
\includegraphics[width=0.82\textwidth]{fig_forest.png}
\caption{Marginal implicit prices of structural attributes (houses, province-FE model M3)
with 95\% cluster-robust confidence intervals, on the log-price scale.}
\label{fig:forest}
\end{figure}

The price--size gradient by dwelling type (Figure~\ref{fig:size_gradient}) confirms the
log--log structure: median price rises concavely with living area for both dwelling types.
Unconditionally, condominiums list \emph{above} houses of the same size---a compositional
effect of their concentration in the expensive metropolitan markets---whereas conditional
on location and ownership the estimated dwelling-type effects show that houses command the
premium, in line with Table~\ref{tab:regression}.

\begin{figure}[t]\centering
\includegraphics[width=0.78\textwidth]{fig_size_gradient.png}
\caption{Median price by living-area bin and dwelling type. The concave gradient is
linearised by the semi-log specification.}
\label{fig:size_gradient}
\end{figure}

\subsection{The geography of housing value}
\label{subsec:geography}

Figure~\ref{fig:maps} maps the spatial structure of value directly. The familiar outline
of populated Canada emerges from the listing coordinates. The Vancouver corridor and the
Greater Toronto--Golden Horseshoe area sit at the top of the price-per-square-metre
distribution, while the Prairies and Atlantic Canada anchor the bottom. The right panel
aggregates to FSA medians, the geographic unit absorbed in the grand model.

\begin{figure}[t]\centering
\begin{subfigure}{0.49\textwidth}\includegraphics[width=\textwidth]{fig_map.png}
\caption{All listings}\end{subfigure}\hfill
\begin{subfigure}{0.49\textwidth}\includegraphics[width=\textwidth]{fig_fsa_map.png}
\caption{FSA neighbourhood medians}\end{subfigure}
\caption{Spatial distribution of housing value. Colour encodes log price per m$^2$; bubble
area in panel~(b) is proportional to $\sqrt{\text{listings}}$. Labels mark the provinces
with substantial samples; the data cover nine provinces, while the territories and Prince
Edward Island contain no listings.}
\label{fig:maps}
\end{figure}

To translate location into a clean dollar statement we recover each FSA's fixed effect
from the grand model---its price premium net of structure, dwelling type and
ownership---and express it relative to the national median (Figure~\ref{fig:premia}). The
highest-valued neighbourhoods, all in the City of Vancouver, trade at \textbf{150--200\%
above} the national-median neighbourhood for an otherwise identical dwelling; the lowest,
in rural Saskatchewan, Manitoba and Newfoundland, sit \textbf{60--67\% below}. The full
premium distribution thus spans a factor of roughly nine between the most and least
expensive neighbourhoods, dwarfing the price range attributable to any single structural
attribute.

\begin{figure}[t]\centering
\includegraphics[width=0.78\textwidth]{fig_premia.png}
\caption{Highest- and lowest-valued neighbourhoods (FSAs) in Canada, measured as the
estimated location premium relative to the national-median neighbourhood, net of
structure, dwelling type and ownership. Only FSAs with at least 50 listings are shown.}
\label{fig:premia}
\end{figure}

\subsection{Decomposing the variance of prices}
\label{subsec:decomp}

Figure~\ref{fig:decomp} casts the specification ladder as a decomposition of the variance
of log prices into the share explained by each successive block of controls. Physical
structure accounts for 46\% of the variance; dwelling type and ownership add a further
0.4~points; province adds about 10~points; and resolving location to the neighbourhood
adds a further 20~points, for a total location contribution near 30~points. Just under a
quarter of the variance remains unexplained and is attributable to idiosyncratic pricing
and unobserved dwelling quality. The visual makes the headline unmistakable: the single
largest identified block of housing value in Canada is the neighbourhood.

\begin{figure}[t]\centering
\includegraphics[width=0.92\textwidth]{fig_decomp.png}
\caption{Variance decomposition of Canadian log house prices into structure, dwelling
type/ownership, province, neighbourhood (FSA) and the unexplained residual, from the
nested specification ladder.}
\label{fig:decomp}
\end{figure}

\subsection{Nonlinearity: diminishing returns to floor space}
\label{subsec:nonlinear}

The constant-elasticity assumption is convenient but restrictive. Re-estimating the grand
model with a quadratic in log living area yields a positive linear term and a
significantly negative quadratic term ($\widehat{\beta}_1=1.06$,
$\widehat{\beta}_2=-0.053$), implying that the marginal elasticity of price with respect
to floor space \emph{declines} with dwelling size. Figure~\ref{fig:nonlinear} traces the
implied marginal elasticity: it falls from roughly $0.65$ for a compact 60~m$^2$ dwelling
to about $0.45$ for a large 350~m$^2$ home. Economically, the first square metres of
living space are valued most highly and additional space is subject to diminishing
returns---consistent with the nonparametric hedonic surfaces of \citet{mcmillen2010issues}.
The constant-elasticity estimate of $0.55$ is best read as an average over the size
distribution.

\begin{figure}[t]\centering
\includegraphics[width=0.78\textwidth]{fig_nonlinear.png}
\caption{Marginal elasticity of price with respect to living area as a function of
dwelling size, from a grand model with a quadratic in log area (95\% cluster-robust band).
The dashed line is the constant-elasticity estimate.}
\label{fig:nonlinear}
\end{figure}

\subsection{The urban price gradient}
\label{subsec:gradient}

Classic urban theory predicts that, holding structure fixed, value declines with distance
from employment centres
\citep{alonso1964location,muth1969cities,mills1967aggregative,ahlfeldt2015density}. We
compute the great-circle distance from each listing to the
nearest of nine major Canadian metropolitan centres and relate it to the location
component of price (the residual from a structure-only model). Figure~\ref{fig:gradient}
confirms a pronounced gradient: dwellings within the metropolitan core carry location
premia of tens of percent over the structure-only benchmark, and the premium decays
steadily with distance, turning negative beyond roughly 80~km. A log-linear fit implies a
semi-elasticity of $-0.085$ ($t>100$): a doubling of distance to the nearest metro is
associated with an 8.5\% lower location premium. This agglomeration gradient is precisely
the force that the FSA fixed effects absorb non-parametrically in the grand model
\citep{combes2015empirics}.

\begin{figure}[t]\centering
\includegraphics[width=0.78\textwidth]{fig_gradient.png}
\caption{Urban price gradient: the location component of price (relative to a
structure-only benchmark) against distance to the nearest of nine major Canadian metros.
The horizontal axis is on a symmetric-log scale.}
\label{fig:gradient}
\end{figure}