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

\subsection{Magnitudes in the light of the literature}

\paragraph{The location share.} Our headline decomposition---structure explains 46\% of
log-price variance, location a further 30 points---is a cross-sectional, listing-level
statement, but its magnitude aligns closely with what aggregate approaches find from the
opposite direction. \citet{davis2007price} estimate that land accounts for roughly 46\%
of the value of the U.S. housing stock, with much higher shares in coastal metros, and
\citet{knoll2017no} attribute about 80\% of the global house-price run-up since 1950 to
land. Land and ``location'' are not identical objects---our FSA premia capitalize
amenities and access as well as physical land scarcity
\citep{cheshire1995price,albouy2016cities}---but both measurements say the same thing:
the dwelling itself is the minority component of housing value. The ×9 span of our
neighbourhood premia likewise echoes the enormous cross-city dispersion of land values
documented by \citet{albouy2016cities} and is exactly the pattern predicted where supply
constraints bind heterogeneously across locations
\citep{saiz2010geographic,gyourko2013superstar}: Vancouver---sea, mountains and an
agricultural land reserve---anchors the top of our premium distribution, as
superstar-city dynamics would predict.

\paragraph{Structural implicit prices.} The living-area elasticity of 0.55 sits inside
the range catalogued by \citet{sirmans2005composition} across decades of published
hedonics, and its decline from 0.66 to 0.55 as location controls tighten is the
classic signature of omitted-locational-quality bias that within-neighbourhood designs
are built to remove \citep{kiel2008location}. The near-zero conditional bedroom
coefficient reproduces one of the most robust findings of that meta-literature: floor
space, not room count, carries value. Our bathroom premium (11--15\%) is at the upper
end of published estimates, which we attribute to bathrooms proxying for unobserved
renovation status and finish quality in listing data that lack an age variable. The
diminishing marginal elasticity of floor space (from $\sim$0.65 at 60~m$^2$ to
$\sim$0.45 at 350~m$^2$) gives parametric confirmation of the curvature that
\citet{mcmillen2010issues} detect nonparametrically.

\paragraph{Spatial dependence.} The 82\% reduction of Moran's~$I$ (0.46 to 0.08) speaks
directly to the oldest empirical worry in housing hedonics
\citep{dubin1988estimation,basu1998analysis}. It quantifies, for a national market, the
claim of \citet{bourassa2007spatial} and \citet{gibbons2012mostly} that geographic
controls---here, three-character postal geography---do most of the work that parametric
spatial-lag structures are designed to do. Our result does not make spatial econometrics
redundant: the residual $I=0.08$ is within-FSA dependence that an explicit
spatio-temporal model \citep{pace1998spatiotemporal,dube2013spatiotemporal} could
exploit, particularly for prediction. It does, however, shift the burden of proof toward
parsimony: a fixed effect per neighbourhood is transparent, imposes no weight-matrix
assumptions, and removes five-sixths of the spatial signal.

\paragraph{Quantile and provincial heterogeneity.} The rising size elasticity across the
price distribution (0.56 at $\tau=0.1$ to 0.60 at $\tau=0.9$) and the rising lot
elasticity mirror the distributional patterns of \citet{zietz2008determinants}, and the
coastal-versus-Prairies gradient in the size elasticity (0.49 in BC to 0.66 in MB) is
consistent with the submarket literature's core claim that implicit prices---not just
price levels---vary across segments \citep{goodman1998housing,bourassa2003submarkets}.
This heterogeneity qualifies our own grand model: the absorbed intercepts allow every
neighbourhood its own price \emph{level}, but the structural slopes are pooled, and
Section~\ref{subsec:heterogeneity} shows those slopes move within an economically
meaningful band. Fully interacted (submarket-specific) coefficient systems are the
natural next step, at a substantial cost in transparency.

\paragraph{Valuation accuracy.} A held-out $R^2$ of 0.764 and a median absolute error of
15.8\% place the transparent hedonic model within the accuracy bands reported in the
mass-appraisal comparison literature \citep{mccluskey2013prediction} and close to the
performance that machine-learning methods deliver on comparable tasks
\citep{mullainathan2017machine}. Two caveats temper the comparison: our target is the
\emph{list} price rather than the transaction price, and our evaluation is restricted to
neighbourhoods observed in training. Within those bounds, the result supports the
position of \citet{clapp2003semiparametric}: most of the predictive content of
``black-box'' valuation lies in the location surface, which a fixed-effect design
captures explicitly and auditably rather than implicitly.

\subsection{Points of tension with prior findings}

Three tensions deserve explicit statement. First, the submarket literature sometimes
finds that \emph{how} submarkets are defined matters little for prediction
\citep{bourassa2003submarkets}; our single-country evidence cannot adjudicate the
optimal geography, and FSAs---postal artifacts---are surely not it. The 0.08 residual
Moran's~$I$ suggests finer geography would still add value. Second,
\citet{kuminoff2010which} warn that hedonic estimates can be fragile to specification;
our robustness battery (six samples, quantiles, quadratic terms) addresses the
first-stage version of this concern, but any second-stage welfare use of our premia
would inherit the well-known identification problems \citep{epple1987hedonic}. Third,
our LOPO exercise shows structural prices transfer imperfectly (mean held-out $R^2$ of
0.36, negative for Alberta), which cuts against reading the grand model's pooled slopes
as universal constants---and aligns with the heterogeneity that segmented-market models
predict \citep{goodman1998housing}.

\subsection{Implications}

For \emph{assessment and property taxation}, the decomposition implies that
neighbourhood-level value---not structure---is where most of the assessable base lives,
so assessment uniformity depends first on getting location surfaces right; the FSA
premia we publish are directly usable as such a surface. For \emph{index construction},
the stability of structural implicit prices across provinces supports pooled hedonic
indices with local intercepts \citep{hill2013hedonic}, while the quantile results
caution that constant-quality adjustment differs across market segments. For
\emph{automated valuation}, the results quantify the price of transparency: a fully
inspectable model concedes little accuracy on within-support predictions, echoing
\citet{mullainathan2017machine}'s point that prediction tasks discipline, rather than
replace, economic structure. For \emph{housing policy}, a ×9 neighbourhood premium span
within one country, concentrated in two metropolitan systems, is the observable imprint
of restricted supply against agglomeration demand
\citep{saiz2010geographic,gyourko2013superstar,combes2015empirics}; policies that expand
supply where premia are highest attack the decomposition's dominant term.

\subsection{Limitations}

Five limitations bound the interpretation. (i)~Prices are \emph{list} prices: sellers
set them strategically \citep{genesove2001loss} and they interact with search and
bargaining \citep{han2015microstructure}, so implicit prices measured on listings can
differ from transaction-based ones, especially in tight markets. (ii)~The data lack year
of construction, renovation status and interior quality; the neighbourhood effects
absorb the cross-neighbourhood component of this unobserved quality, but the within-FSA
structural estimates remain exposed to within-neighbourhood quality sorting.
(iii)~Lot information is sparse and noisily parsed from free text, limiting the
precision of the land-versus-structure margin at the listing level. (iv)~The analysis is
a single cross-section: it characterises the spatial and structural \emph{level} of
prices, not their dynamics, and cannot speak to the efficiency questions of the
repeat-sales tradition \citep{case1989efficiency}. (v)~FSA boundaries are postal
conveniences; premia estimated at this resolution average over genuinely finer
neighbourhood variation, as the residual spatial autocorrelation confirms.

\subsection{Future work}

The natural agenda follows the limitations: linking listings to closing prices and
time-on-market to estimate the list-to-sale wedge \citep{genesove2001loss,han2015microstructure};
enriching the attribute set with age and quality; decomposing the neighbourhood premia
into capitalized amenities---schools, transit, environmental quality---in the
quasi-experimental tradition \citep{black1999better,chay2005does,linden2008estimates};
modelling the residual within-FSA dependence with spatio-temporal structure
\citep{dube2013spatiotemporal}; extending the cross-section to a panel to study
dynamics and policy incidence; and estimating submarket-specific coefficient systems to
map the heterogeneity that our provincial results only sketch
\citep{goodman1998housing,bourassa2003submarkets}.