% ============================================================================
\section{Methodology}
\label{sec:methodology}
% ============================================================================

Our empirical strategy has two parts: a feature-engineering pipeline that turns raw
listings into an analysis-ready design matrix (Sections~\ref{subsec:structural}--%
\ref{subsec:vision}), and a hedonic regression that maps those features to price
(Section~\ref{subsec:econometrics}).

\subsection{Theoretical Framework}
\label{subsec:theory}

Following \citet{rosen1974hedonic}, a house is a bundle of $K$ characteristics
$\mathbf{z}=(z_1,\dots,z_K)$, and the market clears through a hedonic price function
$P(\mathbf{z})$ that equates the marginal bid of buyers to the marginal offer of
sellers. A buyer with income $y$ and preferences $U(x,\mathbf{z})$ over a
numéraire good $x$ and housing attributes chooses $\mathbf{z}$ to solve
%
\begin{equation}
  \max_{x,\,\mathbf{z}}\; U(x,\mathbf{z})
  \quad\text{s.t.}\quad y = x + P(\mathbf{z}),
  \label{eq:consumer}
\end{equation}
%
which yields the first-order condition that, at the optimum, the implicit
(\emph{marginal}) price of each attribute equals the buyer's marginal willingness to
pay:
%
\begin{equation}
  p_k(\mathbf{z}) \;\equiv\; \frac{\partial P(\mathbf{z})}{\partial z_k}
  \;=\; \frac{\partial U/\partial z_k}{\partial U/\partial x},
  \qquad k = 1,\dots,K.
  \label{eq:implicit}
\end{equation}
%
Equation~\eqref{eq:implicit} is the object of interest: the slope of the hedonic
surface in the direction of attribute~$z_k$. The empirical task is to recover
$P(\mathbf{z})$---and hence its gradient---from observed prices and characteristics.
The linear model of Section~\ref{subsec:econometrics} imposes a parametric form on
$P(\cdot)$ and reads the implicit prices off its coefficients; the machine-learning
estimators of Section~\ref{sec:benchmark} approximate $P(\cdot)$ nonparametrically,
trading the closed-form gradient for predictive flexibility. Two caveats apply
throughout. The observed price is an asking price, so $P(\cdot)$ is the hedonic
function of sellers' offers rather than of transactions; and the machine-generated
attributes are measured with error and chosen by the seller, so their coefficients
are predictive associations rather than structural marginal prices
(Section~\ref{subsec:limits}).

\subsection{Structural and Locational Attributes}
\label{subsec:structural}

Structural attributes are extracted by deterministic rules. Living area and lot area
are reported in heterogeneous units (\texttt{"1535 sqft"}, \texttt{"1,333.2 ft²"},
\texttt{"11 561.52 ft²"}, \texttt{"961.6 m2"}); we parse each string and convert all
areas to square feet. For Realtor.ca listings that report room-by-room dimensions
(e.g.\ \texttt{"23 ft ,4 in x 13 ft ,8 in"}) but no aggregate living area, we sum the
parsed room areas and calibrate the total against listings where both are observed;
the median ratio of reported area to summed-room area is~0.98, so the reconstructed
value is used directly to fill missing living area. Age is computed as
$2026-\text{year built}$, entered together with its square to allow a non-monotone
depreciation profile. Bedrooms, bathrooms, parking spaces, and storeys are taken
from the attribute block. Missing values are imputed by the within-property-type
median, and each imputed field is accompanied by a ``\texttt{has\_}'' indicator so
that the regression can distinguish observed from imputed values.

Location is captured by \textbf{municipality fixed effects}. City names are
normalized (Realtor.ca appends neighbourhood qualifiers in parentheses, which we
strip), and municipalities with fewer than 20 listings are pooled into a residual
category, leaving 195 city effects; the pooled group of small municipalities is the
omitted (reference) category, so every reported municipality effect is a premium or
discount relative to a listing in a small municipality. In robustness checks we
replace the municipality effects by \textbf{forward sortation area (RTA) effects},
the first three characters of the postal code, which are available for all but one
listing: 419 RTAs appear in the sample, of which 250 have at least 20 listings and
receive their own effect. Property-type fixed effects and a source indicator complete
the locational and platform controls.

\subsection{Rule-Based Amenities and Room Detail}
\label{subsec:amenities}

Fifteen amenity indicators are detected by bilingual (French/English) keyword
matching over the concatenated feature list, description, and title: swimming pool,
waterfront, garage, fireplace, central air conditioning or heat pump, finished
basement, proximity to water, cul-de-sac, proximity to a golf course, ski hill,
park, school, highway, and public transit, and the absence of rear neighbours. From
the Realtor.ca room block we additionally construct room-detail indicators---presence
of a primary bedroom, a basement, an office, a family room, a workshop, a wine
cellar, a sunroom, a mezzanine, a library, and a laundry room---together with the
room and storey counts. Two listing-quality signals, the number of photographs
(capped at~60) and the presence of a virtual tour, proxy for the marketing tier of
the listing. Finally, following the evidence of Section~\ref{subsec:text_content},
two language indicators (English, mixed; French omitted) and the description length
are available as controls.

\subsection{Language-Model Specifications from Text}
\label{subsec:llm}

Free-text descriptions contain valuation-relevant information that resists keyword
extraction: whether the home is turnkey or a fixer-upper, whether the kitchen or
bathroom has been renovated, the flooring and countertop materials, the heating
system, the presence of a water view, and income potential. We extract these as a
fixed set of structured specifications using a large language model
(\texttt{gpt-4o-mini}, OpenAI, called through the Chat Completions API in July 2026)
with \emph{schema-constrained decoding}: the request specifies a strict JSON schema
with closed enumerations and required fields, so the API validates the output at
generation time and the model returns exactly the expected fields in the expected
value domains (Table~\ref{tab:app_features}). Decoding temperature is set to zero,
and the system instruction, written in French, constrains the model to rely only on
the text and to code absent information as \texttt{false}, \texttt{unknown}, or an
empty list. Each call consumes roughly 600 input and 110 output tokens; the
extraction is run in parallel, is resumable through an on-disk cache keyed by the
listing identifier, and retries transient errors with exponential backoff, for a
total cost of about \$3 across the sample. Descriptions are truncated at 4{,}000
characters, which affects only the longest DuProprio property sheets.

\subsection{Vision Attributes from the Façade Photograph}
\label{subsec:vision}

To capture the visual quality that neither structured fields nor text convey, we
analyze one photograph per listing---the façade---with a vision--language model
(\texttt{claude-haiku-4-5}, Anthropic, Messages API, July 2026). For Realtor.ca the
first image is the exterior shot; for DuProprio, whose images are captioned, we
select the first image whose caption is not a room name. The model returns twelve
fields under a strict JSON schema: an exterior-shot flag, a visual building type, an
overall condition score (1--10), a curb-appeal score (1--10), a renovation level,
the list of exterior materials, landscaping quality, a visible-garage flag, the
visual storey count, the setting (dense urban, suburban, rural, waterfront, wooded),
a photo-quality score, and a luxury indicator. The system instruction directs the
model to ground each score in concrete criteria (surface condition, materials,
landscaping) rather than taste, and to flag non-exterior images. Images are passed
by URL without local download; each call consumes roughly 2{,}470 input and 100
output tokens, for a total cost of about \$57 across the sample. As with the text
layer, the pass is parallelized, resumable, and retried. Unlike the text layer, the
vision calls did not set a sampling temperature and therefore ran at the API default,
so repeated calls could return slightly different scores for borderline images; we
did not re-run the extraction to measure this test--retest variability (see
Section~\ref{subsec:limits}).

\paragraph{From schema fields to regressors.}
Imposing a typed schema with enumerations, required fields, and
\texttt{additionalProperties:false} moves validation into the API layer, so the
outputs are tabulated variables that require no free-text post-processing. They are
then mapped to numeric regressors: booleans to $0/1$; the six-level text condition
scale to an ordinal score $0$--$5$ (\textit{to renovate} $=0$, \ldots, \textit{new}
$=5$, with \textit{unknown} placed at the ``good'' midpoint of~2); the five-level
visual renovation scale to $0$--$4$ and landscaping to $0$--$3$; multi-valued
fields (flooring, heating, exterior materials) and the setting to one indicator per
category. The twelve vision fields thus generate 21 regressors (two scores, the
photo-quality score, two ordinal scales, three indicators, five setting indicators,
and eight material indicators), and the text schema 18 regressors. A listing whose
photograph is not an exterior shot keeps the model's best-effort scores and is
flagged by the exterior indicator. Model identifiers are API aliases that resolve to
the provider's current snapshot; we record the alias and the extraction date, and
the raw JSON outputs are archived so that every regressor can be regenerated from
them without new API calls.

\subsection{Econometric Specification}
\label{subsec:econometrics}

We adopt a semi-logarithmic functional form, the standard workhorse in the hedonic
literature \citep{malpezzi2003hedonic}. Collect the attributes of listing~$i$ into
disjoint blocks,
%
\begin{equation}
  \resizebox{.94\textwidth}{!}{$\displaystyle
  \mathbf{X}_i =
  \big(\underbrace{\ln\text{Area}_i,\ \ln\text{Lot}_i}_{\text{size (logged)}},\
  \underbrace{\mathbf{s}_i}_{\text{structural}},\
  \underbrace{\mathbf{a}_i}_{\text{amenities}},\
  \underbrace{\mathbf{r}_i}_{\text{rooms}},\
  \underbrace{\mathbf{m}_i}_{\text{listing}},\
  \underbrace{\mathbf{g}_i}_{\text{LLM text}},\
  \underbrace{\mathbf{v}_i}_{\text{vision}}\big)
  $}
  \label{eq:blocks}
\end{equation}
%
and estimate, by ordinary least squares,
%
\begin{equation}
  \begin{aligned}
  \ln P_i \;=\;& \alpha
  \;+\; \beta\,\ln\text{Area}_i
  \;+\; \gamma\,\ln\text{Lot}_i
  \;+\; \mathbf{s}_i^{\!\top}\bm{\delta}_s + \mathbf{a}_i^{\!\top}\bm{\delta}_a
  + \mathbf{r}_i^{\!\top}\bm{\delta}_r + \mathbf{m}_i^{\!\top}\bm{\delta}_m \\
  &+ \underbrace{\mathbf{g}_i^{\!\top}\bm{\delta}_g}_{\text{LLM text}}
  + \underbrace{\mathbf{v}_i^{\!\top}\bm{\delta}_v}_{\text{vision}}
  \;+\; \underbrace{\mu_{c(i)}}_{\text{city FE}}
  \;+\; \underbrace{\tau_{t(i)}}_{\text{type FE}}
  \;+\; \varepsilon_i ,
  \end{aligned}
  \label{eq:hedonic}
\end{equation}
%
where $c(i)$ and $t(i)$ index the municipality and property type of listing~$i$, and
$\mu_c$, $\tau_t$ are the corresponding fixed effects that sweep out location- and
type-specific price levels. The source indicator is part of $\mathbf{m}_i$. In
matrix form $\mathbf{y}=\mathbf{Z}\bm{\theta}+\bm{\varepsilon}$ with
$\widehat{\bm{\theta}}=(\mathbf{Z}^{\!\top}\mathbf{Z})^{-1}\mathbf{Z}^{\!\top}\mathbf{y}$.
The contribution of a feature layer is measured by the change in $R^2$ when its
block is added to the specification that contains all preceding blocks; the order
is size and structural, amenities, rooms and listing signals, text, vision, so the
text layer is credited only with what the keyword amenities do not already capture,
and the vision layer only with what neither the text nor the structured fields
capture.

\paragraph{Interpretation of the coefficients.}
Because the size variables enter in logs, $\beta$ and $\gamma$ are \emph{elasticities}:
$\partial \ln P/\partial \ln\text{Area}=\beta$. For an indicator regressor $d$ with
coefficient $b$, the percentage price effect is not $b$ but
%
\begin{equation}
  \%\Delta P \;=\; \big(e^{\,b}-1\big)\times 100,
  \label{eq:halvorsen}
\end{equation}
%
the \citet{halvorsen1980interpretation} correction we apply throughout (for small
$b$, $e^{b}-1\approx b$; the \citealp{kennedy1981estimation} adjustment
$e^{\,b-\frac12\widehat{\mathrm{Var}}(b)}-1$ moves none of our reported effects by
more than 0.1 percentage point). For an ordinal score such as curb appeal, $b$ is the
log-price difference associated with one step of the scale, holding the other
attributes fixed. Under Equation~\eqref{eq:hedonic} the implicit price of a
continuous attribute $z_k$ from \eqref{eq:implicit} is $p_k=\delta_k\,P/z_k$, so
the semi-log form lets the \emph{dollar} marginal price scale with the price level
while keeping the \emph{percentage} effect constant.

\paragraph{Two-part coverage design.}
Because living area is reported for only about 38\% of Realtor.ca listings, we
estimate two nested versions of \eqref{eq:hedonic} and combine their predictions:
%
\begin{equation}
  \widehat{\ln P}_i =
  \begin{cases}
    \widehat{\ln P}_i^{\,A} & \text{if area is measured (Model A, rich),}\\[2pt]
    \widehat{\ln P}_i^{\,B} & \text{otherwise (Model B, broad; }\beta\equiv 0).
  \end{cases}
  \label{eq:twopart}
\end{equation}
%
Model~A includes $\ln\text{Area}$ and is fit on the 9{,}446 listings with a measured
or reconstructed area; Model~B drops it and is fit on the full cleaned sample, so
every listing can be priced.

\paragraph{Inference.}
House-price residuals are strongly heteroskedastic, and listings within a
municipality share unobserved local shocks that induce within-cluster error
correlation. Our preferred standard errors are therefore clustered at the
municipality level (195 clusters), the same level as the fixed effects
\citep{cameron2015cluster,abadie2023when}:
%
\begin{equation}
  \widehat{\mathrm{Var}}_{\text{CL}}(\widehat{\bm{\theta}})
  = (\mathbf{Z}^{\!\top}\mathbf{Z})^{-1}
    \Bigg(\sum_{c=1}^{C}
    \mathbf{Z}_c^{\!\top}\hat{\bm{\varepsilon}}_c
    \hat{\bm{\varepsilon}}_c^{\!\top}\mathbf{Z}_c\Bigg)
    (\mathbf{Z}^{\!\top}\mathbf{Z})^{-1},
  \label{eq:cluster}
\end{equation}
%
where $\mathbf{Z}_c$ and $\hat{\bm{\varepsilon}}_c$ stack the regressors and
residuals of cluster $c$. Cluster sizes are very unequal---Montreal alone contributes
about one listing in twenty, and the pooled residual category of small municipalities,
which is treated as a single cluster, holds about one listing in four (5{,}006 listings from 999 municipalities)---a configuration in which the asymptotic cluster-robust
variance can be unreliable \citep{mackinnon2017wild}. We therefore also report a
pairs cluster bootstrap that resamples municipalities with replacement
($B=299$) and takes the 2.5th and 97.5th percentiles of the resampled coefficients
as a confidence interval. As a complement we compute heteroskedasticity-consistent
(HC3) standard errors \citep{white1980heteroskedasticity,mackinnon1985some};
Section~\ref{subsec:robust} shows that the three approaches agree on the
significance of the property-level AI attributes and differ mainly for keyword
amenities with a strong spatial component. When RTA fixed effects replace
municipality effects, standard errors are clustered by RTA.

\paragraph{Retransformation.}
Predicting the \emph{level} of price requires undoing the log. Because
$\mathbb{E}[P\mid\mathbf{Z}]=\exp(\mathbf{Z}\bm{\theta})\,\mathbb{E}[e^{\varepsilon}]
\neq \exp(\mathbf{Z}\bm{\theta})$, naive exponentiation is biased downward. Under
$\varepsilon\sim\mathcal N(0,\sigma^2)$ we use the closed-form correction, and more
generally Duan's nonparametric smearing estimate \citep{duan1983smearing}:
%
\begin{equation}
  \widehat{P}_i
  = \exp\!\big(\widehat{\ln P}_i\big)\cdot
  \underbrace{\exp\!\big(\tfrac12\hat\sigma^{2}\big)}_{\text{normal}}
  \qquad\text{or}\qquad
  \widehat{P}_i
  = \exp\!\big(\widehat{\ln P}_i\big)\cdot
  \underbrace{\frac1n\sum_{j=1}^{n} e^{\hat\varepsilon_j}}_{\text{smearing }\hat\phi}.
  \label{eq:smearing}
\end{equation}
%
The out-of-sample accuracy metrics of Section~\ref{sec:benchmark} are computed on
$\ln P$ (for $R^2$) and on the ratio of predicted to actual price (for the median
absolute percentage error), neither of which depends on the retransformation
constant.

\paragraph{Out-of-sample validation.}
All predictive comparisons use five-fold cross-validation with a common seed, so that
every estimator and every feature set is evaluated on identical held-out folds. Under
\emph{random} folds a held-out house typically has close neighbours in the training
folds, which flatters any model that can exploit spatial proximity
\citep{roberts2017cross}. We therefore repeat the exercise with \emph{spatially
grouped} folds, assigning entire RTAs to folds so that no held-out listing shares
its RTA with a training listing; because many small municipalities consist of a
single RTA, their municipality indicator is then unobserved in training and the
model prices them at the reference level. The spatial design is a deliberately harsh
test of how the estimators and feature layers behave for a new area; the random
design corresponds to the operational use of an AVM that prices a new listing in a
market it already covers.