spb/wp9_uqo Public

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Empirical Strategy}4\label{sec:methodology}5% ============================================================================67\subsection{The hedonic equation}8\label{subsec:model}910We estimate the semi-logarithmic hedonic price equation11%12\begin{equation}13  \ln P_{i}14  = \alpha15  + \beta\,\ln(\text{Area}_{i})16  + \mathbf{x}_{i}'\boldsymbol{\gamma}17  + \mathbf{d}_{i}'\boldsymbol{\delta}18  + \mu_{f(i)}19  + \varepsilon_{i},20  \label{eq:hedonic}21\end{equation}22%23where $P_{i}$ is the list price of dwelling~$i$; $\text{Area}_{i}$ is living area in24square metres; $\mathbf{x}_{i}$ collects the remaining structural attributes (bedrooms,25full and half bathrooms, parking, storeys, the lot indicator and $\ln$ lot area);26$\mathbf{d}_{i}$ is a vector of dwelling-type and ownership dummies; $\mu_{f(i)}$ is a27fixed effect for the FSA neighbourhood $f$ to which dwelling~$i$ belongs; and28$\varepsilon_{i}$ is an idiosyncratic error.2930Because $\beta$ multiplies a logged regressor, it is the \emph{elasticity} of price with31respect to floor space. The elements of $\boldsymbol{\gamma}$ attached to count variables32are semi-elasticities: following \citet{halvorsen1980interpretation}, a coefficient33$\gamma$ implies an approximate proportional price change of $100\,(e^{\gamma}-1)\%$ for a34one-unit increase, which we report whenever the distinction from the raw coefficient is35material.3637\subsection{High-dimensional location effects}38\label{subsec:identification}3940The neighbourhood fixed effects $\mu_{f}$ are the heart of the design. By including a41separate intercept for each of the 1{,}153 FSAs, we allow every neighbourhood an arbitrary42price level that absorbs all location-specific amenities---school quality, transit access,43coastline, employment density---whether or not they are observed. The structural implicit44prices in $\beta$ and $\boldsymbol{\gamma}$ are then identified purely from variation45\emph{within} neighbourhoods, comparing dwellings that differ in their physical attributes46but share a location. This is the housing counterpart of the within estimator, and it47addresses the most pernicious source of omitted-variable bias in hedonic work: the48correlation between structural quality and unobserved locational quality49\citep{can1992specification,bourassa2007spatial}.5051Including more than a thousand dummies directly is numerically wasteful. We instead52estimate Equation~\eqref{eq:hedonic} by \emph{absorbing least squares}, in the53high-dimensional fixed-effects tradition initiated by \citet{abowd1999high}: the FSA54effects are partialled out of both the dependent variable and the regressors before the55structural coefficients are estimated \citep{guimaraes2010simple,correia2017reghdfe},56and the slope estimates are numerically identical to full-dummy OLS. All standard errors57are clustered at the FSA level to allow for arbitrary within-neighbourhood correlation58and heteroskedasticity \citep{cameron2015practitioner}.5960\subsection{Specification ladder}61\label{subsec:ladder}6263We report a ladder of nested specifications that isolates the marginal contribution of64each block of controls:65%66\begin{itemize}67  \item \textbf{M1 -- Structural:} structural attributes only.68  \item \textbf{M2 -- + Type/Ownership:} adds dwelling-type and ownership dummies.69  \item \textbf{M3 -- + Province:} adds province fixed effects.70  \item \textbf{M4 -- Houses + FSA:} replaces province with FSA fixed effects, houses only.71  \item \textbf{M5 -- Grand model:} FSA fixed effects over all residential dwellings.72\end{itemize}73%74Columns M1--M3 are estimated on the house subsample to keep the structural interpretation75clean; M5 is the preferred grand specification estimated over the full residential sample.76The gap in $R^2$ between M3 and M5 measures the explanatory value of resolving location at77the neighbourhood rather than the provincial scale.7879\subsection{Retransformation and out-of-sample evaluation}80\label{subsec:retrans}8182Because the model is estimated in logs, predicted price levels require a retransformation83correction. We use Duan's smearing estimator \citep{duan1983smearing}, multiplying84$\exp(\widehat{\ln P})$ by the sample mean of $\exp(\widehat{\varepsilon})$, which is85consistent without assuming log-normal errors. For predictive validation we randomly split86the data 80/20, estimate the model on the training fold (restricting evaluation to87neighbourhoods observed in training, since out-of-support FSA effects are not identified),88and report held-out fit and percentage-error metrics on the test fold.89