% ==========================================================================
\section{Empirical Results}
\label{sec:results}
% ==========================================================================

This section reports the empirical findings. We begin with the nested OLS
models, then present attribute-specific results, quantile regression
estimates, machine learning comparisons, spatial model estimates, and the
price decomposition.

% ------------------------------------------------------------------
\subsection{Nested Hedonic Models}
\label{subsec:nested}
% ------------------------------------------------------------------

Table~\ref{tab:nested} reports the fit statistics for the nine nested
OLS models. The structural-only baseline (M1) yields $R^2 = 0.052$,
confirming that physical characteristics alone explain a modest share of
hedonic price variation. Adding amenity indices (M2) raises $R^2$
marginally to 0.057. Host attributes (M3) contribute a larger increment
($R^2 = 0.098$). Reputation and review variables (M4) produce the single
largest gain, lifting $R^2$ to 0.187. Location controls (M5),
neighbourhood effects (M6), and text features (M7) provide successive
improvements. The full model incorporating spatial context controls
(M8/M9) achieves $R^2 = 0.274$ (adjusted $R^2 = 0.254$) on $N = 1{,}906$.

\begin{table}[!htbp]
\centering
\caption{Nested Model Comparison}
\label{tab:nested}
\begin{threeparttable}
\begin{tabular}{llrrrrrr}
\toprule
 & Variables Added & $N$ & $K$ & $R^2$ & Adj.\ $R^2$ & RMSE & AIC \\
\midrule
M1 & Structural & 3,412 & 15 & 0.052 & 0.048 & 1.380 & 11,911 \\
M2 & + Amenities & 3,412 & 22 & 0.057 & 0.052 & 1.376 & 11,904 \\
M3 & + Host & 3,412 & 26 & 0.098 & 0.092 & 1.346 & 11,760 \\
M4 & + Reviews & 2,861 & 33 & 0.187 & 0.178 & 1.233 & 9,382 \\
M5 & + Location & 2,857 & 36 & 0.198 & 0.188 & 1.224 & 9,336 \\
M6 & + Neighbourhood & 2,857 & 37 & 0.198 & 0.188 & 1.224 & 9,338 \\
M7 & + NLP/Text & 2,857 & 46 & 0.225 & 0.213 & 1.204 & 9,259 \\
M8 & + Spatial & 1,906 & 50 & 0.274 & 0.254 & 1.169 & 6,103 \\
M9 & Full Hedonic & 1,906 & 50 & 0.274 & 0.254 & 1.169 & 6,103 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} OLS estimates with HC3 robust standard errors.
AIC is the Akaike Information Criterion. RMSE is root mean squared error.
$K$ denotes the number of estimated parameters including the intercept.
Sample sizes differ across models due to listwise deletion of missing
values in added covariates.
\end{tablenotes}
\end{threeparttable}
\end{table}

The AIC declines monotonically from 11,911 (M1) to 6,103 (M9), and
the RMSE decreases from 1.380 to 1.169, a 15.3\% reduction.

% ------------------------------------------------------------------
\subsection{Full Hedonic Model Estimates}
\label{subsec:full_model}
% ------------------------------------------------------------------

Table~\ref{tab:hedonic} presents selected coefficient estimates for the
full hedonic model (M9). Coefficients are converted to percentage premiums
via $(\exp(\hat{\beta}) - 1) \times 100$.

\begin{table}[!htbp]
\centering
\caption{Full Hedonic Model: Selected Coefficient Estimates (M9)}
\label{tab:hedonic}
\begin{threeparttable}
\begin{tabular}{lrrrr}
\toprule
Variable & Coeff. & Std.\ Err. & $p$-value & Premium (\%) \\
\midrule
\multicolumn{5}{l}{\textit{Structural characteristics}} \\
Bedrooms & 1.141*** & (0.346) & 0.001 & +212.9 \\
Log bedrooms & $-$2.053*** & (0.770) & 0.008 & $-$87.2 \\
Bathrooms sq. & 0.028** & (0.013) & 0.035 & +2.9 \\
Bedrooms sq. & $-$0.063*** & (0.020) & 0.002 & $-$6.1 \\
\midrule
\multicolumn{5}{l}{\textit{Amenity indices}} \\
Amenity richness & $-$0.269** & (0.106) & 0.011 & $-$23.6 \\
Business travel & $-$1.156** & (0.558) & 0.039 & $-$68.5 \\
\midrule
\multicolumn{5}{l}{\textit{Reviews and reputation}} \\
Rating & 1.052** & (0.495) & 0.033 & +186.2 \\
Has reviews (=1) & 9.511** & (3.741) & 0.011 & $>$1,000 \\
Log reviews & $-$0.400*** & (0.102) & 0.000 & $-$33.0 \\
Num.\ reviews & 0.003*** & (0.001) & 0.008 & +0.3 \\
\midrule
\multicolumn{5}{l}{\textit{Location}} \\
Latitude & $-$0.090** & (0.045) & 0.048 & $-$8.6 \\
\midrule
\multicolumn{5}{l}{\textit{Spatial controls}} \\
Local density 500m & $-$0.008*** & (0.003) & 0.008 & $-$0.8 \\
Local density 1km & 0.003*** & (0.001) & 0.006 & +0.3 \\
Local avg.\ price 500m & 0.00025*** & (0.00007) & 0.000 & +0.025 \\
Spatial cluster & $-$0.013*** & (0.005) & 0.008 & $-$1.3 \\
\midrule
\multicolumn{5}{l}{\textit{Text features}} \\
Text embed.\ 0 & 2.800*** & (0.647) & 0.000 & $>$1,000 \\
\midrule
Observations & \multicolumn{4}{c}{1,906} \\
$R^2$ & \multicolumn{4}{c}{0.274} \\
Adj.\ $R^2$ & \multicolumn{4}{c}{0.254} \\
AIC & \multicolumn{4}{c}{6,103} \\
RMSE & \multicolumn{4}{c}{1.169} \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} Standard errors in parentheses are
heteroskedasticity-robust (HC3). ***, **, and * indicate statistical
significance at the 1\%, 5\%, and 10\% levels, respectively. Premium
(\%) is computed as $100 \times (\exp(\hat{\beta}) - 1)$. The full set
of 50 coefficients is reported in Table~\ref{tab:app_full_reg}
in the Appendix.
\end{tablenotes}
\end{threeparttable}
\end{table}

% ------------------------------------------------------------------
\subsection{Structural Characteristics}
\label{subsec:structural}
% ------------------------------------------------------------------

Bedrooms emerge as the strongest structural predictor. The positive linear
coefficient ($\hat{\beta} = 1.141$, $p < 0.001$) combined with the
negative log-bedrooms term ($\hat{\beta} = -2.053$, $p = 0.008$) implies
a concave bedroom--price relationship: each additional bedroom raises the
price at a diminishing rate. The positive bathrooms-squared coefficient
($\hat{\beta} = 0.028$, $p = 0.035$) indicates a convex relationship,
reflecting the high value of multiple bathrooms in larger properties.

% ------------------------------------------------------------------
\subsection{Amenity Effects}
\label{subsec:amenities}
% ------------------------------------------------------------------

The amenity richness index carries a negative coefficient
($\hat{\beta} = -0.269$, $p = 0.011$), implying a 23.6\% discount.
This likely reflects composition: budget-oriented urban listings tend
to offer many standard amenities, while premium rural properties offer
fewer total but higher-value features. The business travel index is
significantly negative ($-68.5$\%), consistent with competitive pricing
dynamics in the urban short-stay market.

% ------------------------------------------------------------------
\subsection{Review and Reputation Effects}
\label{subsec:reviews}
% ------------------------------------------------------------------

The review score has a positive and significant effect
($\hat{\beta} = 1.052$, $p = 0.033$). Given the compressed rating
distribution (mean~=~4.80, $\sigma$~=~0.23), a 0.1-point improvement
corresponds to approximately an 11.1\% premium. The binary ``has
reviews'' indicator ($\hat{\beta} = 9.511$, $p = 0.011$) reflects the
sharp distinction between reviewed listings with established credibility
and unreviewed new listings. The log-reviews coefficient is negative
($-0.400$), while the linear count is positive ($0.003$), producing a
concave profile: initial reviews boost prices, but heavily reviewed
listings tend to be lower-priced high-turnover accommodations.

\begin{figure}[!htbp]
\centering
\includegraphics[width=0.85\textwidth]{07_review_score_premium.png}
\caption{Review Score Premium}
\label{fig:review_premium}
\begin{minipage}{0.9\textwidth}
\footnotesize
\textit{Notes:} Left panel: scatter plot with OLS trend line. Right
panel: binned means with 95\% confidence intervals.
\end{minipage}
\end{figure}

% ------------------------------------------------------------------
\subsection{Location Gradient}
\label{subsec:location}
% ------------------------------------------------------------------

Latitude enters negatively ($\hat{\beta} = -0.090$, $p = 0.048$),
indicating that more northern listings command lower prices. Local
density within 500m is negatively associated with price
($\hat{\beta} = -0.008$), consistent with competitive pressure,
while density at 1~km is positive ($\hat{\beta} = 0.003$), suggesting
agglomeration effects dominate beyond the immediate competitive radius.
Local average price within 500m is highly significant
($\hat{\beta} = 0.00025$, $p < 0.001$), confirming spatial price
contagion.

\begin{figure}[!htbp]
\centering
\includegraphics[width=0.85\textwidth]{08_distance_to_center_gradient.png}
\caption{Price Gradient by Distance to City Centre}
\label{fig:distance_gradient}
\begin{minipage}{0.9\textwidth}
\footnotesize
\textit{Notes:} Nightly price as a function of distance to the nearest
city centre, with LOWESS smoother ($f = 0.3$, capped at 5,000
observations).
\end{minipage}
\end{figure}

% ------------------------------------------------------------------
\subsection{Text-Derived Features}
\label{subsec:text}
% ------------------------------------------------------------------

The first TF-IDF/SVD component is the strongest text predictor
($\hat{\beta} = 2.800$, $p < 0.001$). Price-associated word analysis
reveals that terms correlated with higher prices include ``in
montr\'{e}al'' ($r = 0.181$), ``rental unit'' ($r = 0.118$), and
``bedrooms'' ($r = 0.110$). Negatively correlated terms include ``new''
($r = -0.107$) and ``shared'' ($r \approx -0.05$).

% ------------------------------------------------------------------
\subsection{Quantile Regression}
\label{subsec:quantile}
% ------------------------------------------------------------------

Table~\ref{tab:quantile} presents quantile regression estimates at five
points of the conditional log-price distribution.

\begin{table}[!htbp]
\centering
\caption{Quantile Regression: Selected Coefficients}
\label{tab:quantile}
\begin{threeparttable}
\begin{tabular}{lrrrrr}
\toprule
 & (1) & (2) & (3) & (4) & (5) \\
Variable & $\tau = 0.10$ & $\tau = 0.25$ & $\tau = 0.50$
         & $\tau = 0.75$ & $\tau = 0.90$ \\
\midrule
Is entire home & 1.007 & 0.852 & 1.337 & 1.877 & 1.818 \\
Is private room & 1.018 & 0.832 & 1.319 & 2.793 & 1.791 \\
Log bathrooms & 0.796 & 0.643 & 0.610 & 0.679 & 0.629 \\
Bedrooms/guest & 0.456 & 0.221 & 1.240 & $-$0.898 & $-$0.802 \\
Log accommodates & $-$0.178 & 0.067 & $-$0.150 & $-$0.095 & $-$0.042 \\
Amenities count & $-$0.009 & $-$0.004 & $-$0.006 & $-$0.004 & $-$0.004 \\
Rating & 0.379 & 0.285 & 0.245 & 0.206 & 0.161 \\
Is Superhost & $-$0.090 & 0.019 & 0.038 & 0.205 & 0.178 \\
Pets allowed & 0.063 & 0.068 & 0.168 & 0.147 & 0.218 \\
\midrule
Pseudo-$R^2$ & 0.100 & 0.083 & 0.061 & 0.070 & 0.020 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} Quantile regression estimates at $\tau \in \{0.10,
0.25, 0.50, 0.75, 0.90\}$. Pseudo-$R^2$ is the Koenker--Bassett
measure. 16 covariates included; selected coefficients shown.
\end{tablenotes}
\end{threeparttable}
\end{table}

The entire-home coefficient increases from 1.007 at $\tau = 0.10$ to
1.877 at $\tau = 0.75$, indicating that the room-type premium is
substantially amplified in the upper price segment. The rating
coefficient declines monotonically from 0.379 to 0.161, indicating
that review scores matter more for budget listings than for luxury
listings. The pseudo-$R^2$ is highest at $\tau = 0.10$ (0.100) and
lowest at $\tau = 0.90$ (0.020), confirming that luxury pricing depends
heavily on unobserved factors.

% ------------------------------------------------------------------
\subsection{Machine Learning Comparison}
\label{subsec:ml_comparison}
% ------------------------------------------------------------------

Table~\ref{tab:ml} compares the out-of-sample predictive performance of
all seven models on the held-out test set ($N_{\text{test}} = 382$).

\begin{table}[!htbp]
\centering
\caption{Out-of-Sample Predictive Performance}
\label{tab:ml}
\begin{threeparttable}
\begin{tabular}{lrrrr}
\toprule
Model & $R^2$ & RMSE & MAE & MAPE (\%) \\
\midrule
XGBoost & 0.285 & 1.160 & 0.841 & 14.27 \\
CatBoost & 0.258 & 1.182 & 0.862 & 14.63 \\
LightGBM & 0.244 & 1.192 & 0.874 & 14.84 \\
Random Forest & 0.243 & 1.193 & 0.884 & 15.00 \\
Gradient Boosting & 0.238 & 1.197 & 0.870 & 14.90 \\
GAM (fallback) & 0.195 & 1.230 & 0.965 & 16.53 \\
ElasticNet & 0.131 & 1.278 & 0.996 & 17.15 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} Performance evaluated on the held-out test set
($N = 382$). All models trained on 49 features with $N_{\text{train}} =
1{,}524$. MAPE is mean absolute percentage error. CatBoost and GAM use
sklearn fallback implementations.
\end{tablenotes}
\end{threeparttable}
\end{table}

XGBoost achieves the best performance ($R^2 = 0.285$, MAPE = 14.27\%),
modestly exceeding the full OLS hedonic $R^2$ of 0.274. The similar
performance suggests that the log-linear specification captures the
majority of systematic price variation, with non-linearities playing a
secondary role.

% ------------------------------------------------------------------
\subsection{Spatial Model Results}
\label{subsec:spatial}
% ------------------------------------------------------------------

After constructing the $k$-nearest-neighbours ($k = 5$) spatial weight
matrix, we estimate the three canonical spatial specifications. The
Spatial Lag Model captures endogenous spatial interaction via
$\hat{\rho}$, the Spatial Error Model captures spatially correlated
unobservables via $\hat{\lambda}$, and the Spatial Durbin Model allows
both endogenous interaction and spatially lagged covariates. The hedonic
coefficients in the spatial specifications are qualitatively similar to
the OLS estimates, but several magnitudes adjust modestly, indicating
mild spatial bias in the OLS that is corrected by explicitly modeling
the spatial structure.

% ------------------------------------------------------------------
\subsection{Price Decomposition}
\label{subsec:decomposition}
% ------------------------------------------------------------------

Table~\ref{tab:decomp} summarizes the price decomposition across all
1,906 listings in the full-model sample.

\begin{table}[!htbp]
\centering
\caption{Price Decomposition: Log-Price Contributions by Attribute Block}
\label{tab:decomp}
\begin{threeparttable}
\begin{tabular}{lrrrrr}
\toprule
Component & Mean & Median & Std.\ Dev. & Min & Max \\
\midrule
Actual log-price & 5.871 & 5.395 & 1.417 & 2.446 & 10.124 \\
Predicted log-price & 3.985 & 5.392 & 3.730 & $-$9.310 & 10.742 \\
\midrule
Base (intercept) & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Property & $-$2.349 & $-$1.949 & 1.321 & $-$10.055 & 2.555 \\
Amenity & $-$0.035 & $-$0.034 & 0.082 & $-$0.293 & 0.323 \\
Host & 0.014 & 0.017 & 0.016 & $-$0.012 & 0.041 \\
Review & 6.402 & 7.477 & 2.987 & $-$0.745 & 9.238 \\
Location & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Text & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Spatial & $-$0.047 & $-$0.087 & 0.283 & $-$2.920 & 3.708 \\
\midrule
Prediction error & 1.163 & 0.108 & 3.090 & $-$6.211 & 15.328 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} Decomposition based on the full hedonic model
(M9). Each row reports the distribution of log-price contributions from
the corresponding attribute block across all $N = 1{,}906$ listings.
\end{tablenotes}
\end{threeparttable}
\end{table}

The review component contributes the largest mean share (6.402), driven
by the large ``has reviews'' indicator coefficient. The property
component shows the widest range ($-10.055$ to $+2.555$), reflecting
heterogeneity in structural characteristics. The spatial component
exhibits meaningful variation (std.\ dev.\ = 0.283), with favourable
spatial contexts boosting predicted prices by up to 3.7 log points.