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

\subsection{The hedonic equation}
\label{subsec:model}

We estimate the semi-logarithmic hedonic price equation
%
\begin{equation}
  \ln P_{i}
  = \alpha
  + \beta\,\ln(\text{Area}_{i})
  + \mathbf{x}_{i}'\boldsymbol{\gamma}
  + \mathbf{d}_{i}'\boldsymbol{\delta}
  + \mu_{f(i)}
  + \varepsilon_{i},
  \label{eq:hedonic}
\end{equation}
%
where $P_{i}$ is the list price of dwelling~$i$; $\text{Area}_{i}$ is living area in
square metres; $\mathbf{x}_{i}$ collects the remaining structural attributes (bedrooms,
full and half bathrooms, parking, storeys, the lot indicator and $\ln$ lot area);
$\mathbf{d}_{i}$ is a vector of dwelling-type and ownership dummies; $\mu_{f(i)}$ is a
fixed effect for the FSA neighbourhood $f$ to which dwelling~$i$ belongs; and
$\varepsilon_{i}$ is an idiosyncratic error.

Because $\beta$ multiplies a logged regressor, it is the \emph{elasticity} of price with
respect to floor space. The elements of $\boldsymbol{\gamma}$ attached to count variables
are semi-elasticities: following \citet{halvorsen1980interpretation}, a coefficient
$\gamma$ implies an approximate proportional price change of $100\,(e^{\gamma}-1)\%$ for a
one-unit increase, which we report whenever the distinction from the raw coefficient is
material.

\subsection{High-dimensional location effects}
\label{subsec:identification}

The neighbourhood fixed effects $\mu_{f}$ are the heart of the design. By including a
separate intercept for each of the 1{,}153 FSAs, we allow every neighbourhood an arbitrary
price level that absorbs all location-specific amenities---school quality, transit access,
coastline, employment density---whether or not they are observed. The structural implicit
prices in $\beta$ and $\boldsymbol{\gamma}$ are then identified purely from variation
\emph{within} neighbourhoods, comparing dwellings that differ in their physical attributes
but share a location. This is the housing counterpart of the within estimator, and it
addresses the most pernicious source of omitted-variable bias in hedonic work: the
correlation between structural quality and unobserved locational quality
\citep{can1992specification,bourassa2007spatial}.

Including more than a thousand dummies directly is numerically wasteful. We instead
estimate Equation~\eqref{eq:hedonic} by \emph{absorbing least squares}, in the
high-dimensional fixed-effects tradition initiated by \citet{abowd1999high}: the FSA
effects are partialled out of both the dependent variable and the regressors before the
structural coefficients are estimated \citep{guimaraes2010simple,correia2017reghdfe},
and the slope estimates are numerically identical to full-dummy OLS. All standard errors
are clustered at the FSA level to allow for arbitrary within-neighbourhood correlation
and heteroskedasticity \citep{cameron2015practitioner}.

\subsection{Specification ladder}
\label{subsec:ladder}

We report a ladder of nested specifications that isolates the marginal contribution of
each block of controls:
%
\begin{itemize}
  \item \textbf{M1 -- Structural:} structural attributes only.
  \item \textbf{M2 -- + Type/Ownership:} adds dwelling-type and ownership dummies.
  \item \textbf{M3 -- + Province:} adds province fixed effects.
  \item \textbf{M4 -- Houses + FSA:} replaces province with FSA fixed effects, houses only.
  \item \textbf{M5 -- Grand model:} FSA fixed effects over all residential dwellings.
\end{itemize}
%
Columns M1--M3 are estimated on the house subsample to keep the structural interpretation
clean; M5 is the preferred grand specification estimated over the full residential sample.
The gap in $R^2$ between M3 and M5 measures the explanatory value of resolving location at
the neighbourhood rather than the provincial scale.

\subsection{Retransformation and out-of-sample evaluation}
\label{subsec:retrans}

Because the model is estimated in logs, predicted price levels require a retransformation
correction. We use Duan's smearing estimator \citep{duan1983smearing}, multiplying
$\exp(\widehat{\ln P})$ by the sample mean of $\exp(\widehat{\varepsilon})$, which is
consistent without assuming log-normal errors. For predictive validation we randomly split
the data 80/20, estimate the model on the training fold (restricting evaluation to
neighbourhoods observed in training, since out-of-support FSA effects are not identified),
and report held-out fit and percentage-error metrics on the test fold.
