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1% ==========================================================================2\section{Empirical Results}3\label{sec:results}4% ==========================================================================56This section reports the empirical findings. We begin with the nested OLS7models, then present attribute-specific results, quantile regression8estimates, machine learning comparisons, spatial model estimates, and the9price decomposition.1011% ------------------------------------------------------------------12\subsection{Nested Hedonic Models}13\label{subsec:nested}14% ------------------------------------------------------------------1516Table~\ref{tab:nested} reports the fit statistics for the nine nested17OLS models. The structural-only baseline (M1) yields $R^2 = 0.052$,18confirming that physical characteristics alone explain a modest share of19hedonic price variation. Adding amenity indices (M2) raises $R^2$20marginally to 0.057. Host attributes (M3) contribute a larger increment21($R^2 = 0.098$). Reputation and review variables (M4) produce the single22largest gain, lifting $R^2$ to 0.187. Location controls (M5),23neighbourhood effects (M6), and text features (M7) provide successive24improvements. The full model incorporating spatial context controls25(M8/M9) achieves $R^2 = 0.274$ (adjusted $R^2 = 0.254$) on $N = 1{,}906$.2627\begin{table}[!htbp]28\centering29\caption{Nested Model Comparison}30\label{tab:nested}31\begin{threeparttable}32\begin{tabular}{llrrrrrr}33\toprule34 & Variables Added & $N$ & $K$ & $R^2$ & Adj.\ $R^2$ & RMSE & AIC \\35\midrule36M1 & Structural & 3,412 & 15 & 0.052 & 0.048 & 1.380 & 11,911 \\37M2 & + Amenities & 3,412 & 22 & 0.057 & 0.052 & 1.376 & 11,904 \\38M3 & + Host & 3,412 & 26 & 0.098 & 0.092 & 1.346 & 11,760 \\39M4 & + Reviews & 2,861 & 33 & 0.187 & 0.178 & 1.233 & 9,382 \\40M5 & + Location & 2,857 & 36 & 0.198 & 0.188 & 1.224 & 9,336 \\41M6 & + Neighbourhood & 2,857 & 37 & 0.198 & 0.188 & 1.224 & 9,338 \\42M7 & + NLP/Text & 2,857 & 46 & 0.225 & 0.213 & 1.204 & 9,259 \\43M8 & + Spatial & 1,906 & 50 & 0.274 & 0.254 & 1.169 & 6,103 \\44M9 & Full Hedonic & 1,906 & 50 & 0.274 & 0.254 & 1.169 & 6,103 \\45\bottomrule46\end{tabular}47\begin{tablenotes}48\footnotesize49\item \textit{Notes:} OLS estimates with HC3 robust standard errors.50AIC is the Akaike Information Criterion. RMSE is root mean squared error.51$K$ denotes the number of estimated parameters including the intercept.52Sample sizes differ across models due to listwise deletion of missing53values in added covariates.54\end{tablenotes}55\end{threeparttable}56\end{table}5758The AIC declines monotonically from 11,911 (M1) to 6,103 (M9), and59the RMSE decreases from 1.380 to 1.169, a 15.3\% reduction.6061% ------------------------------------------------------------------62\subsection{Full Hedonic Model Estimates}63\label{subsec:full_model}64% ------------------------------------------------------------------6566Table~\ref{tab:hedonic} presents selected coefficient estimates for the67full hedonic model (M9). Coefficients are converted to percentage premiums68via $(\exp(\hat{\beta}) - 1) \times 100$.6970\begin{table}[!htbp]71\centering72\caption{Full Hedonic Model: Selected Coefficient Estimates (M9)}73\label{tab:hedonic}74\begin{threeparttable}75\begin{tabular}{lrrrr}76\toprule77Variable & Coeff. & Std.\ Err. & $p$-value & Premium (\%) \\78\midrule79\multicolumn{5}{l}{\textit{Structural characteristics}} \\80Bedrooms & 1.141*** & (0.346) & 0.001 & +212.9 \\81Log bedrooms & $-$2.053*** & (0.770) & 0.008 & $-$87.2 \\82Bathrooms sq. & 0.028** & (0.013) & 0.035 & +2.9 \\83Bedrooms sq. & $-$0.063*** & (0.020) & 0.002 & $-$6.1 \\84\midrule85\multicolumn{5}{l}{\textit{Amenity indices}} \\86Amenity richness & $-$0.269** & (0.106) & 0.011 & $-$23.6 \\87Business travel & $-$1.156** & (0.558) & 0.039 & $-$68.5 \\88\midrule89\multicolumn{5}{l}{\textit{Reviews and reputation}} \\90Rating & 1.052** & (0.495) & 0.033 & +186.2 \\91Has reviews (=1) & 9.511** & (3.741) & 0.011 & $>$1,000 \\92Log reviews & $-$0.400*** & (0.102) & 0.000 & $-$33.0 \\93Num.\ reviews & 0.003*** & (0.001) & 0.008 & +0.3 \\94\midrule95\multicolumn{5}{l}{\textit{Location}} \\96Latitude & $-$0.090** & (0.045) & 0.048 & $-$8.6 \\97\midrule98\multicolumn{5}{l}{\textit{Spatial controls}} \\99Local density 500m & $-$0.008*** & (0.003) & 0.008 & $-$0.8 \\100Local density 1km & 0.003*** & (0.001) & 0.006 & +0.3 \\101Local avg.\ price 500m & 0.00025*** & (0.00007) & 0.000 & +0.025 \\102Spatial cluster & $-$0.013*** & (0.005) & 0.008 & $-$1.3 \\103\midrule104\multicolumn{5}{l}{\textit{Text features}} \\105Text embed.\ 0 & 2.800*** & (0.647) & 0.000 & $>$1,000 \\106\midrule107Observations & \multicolumn{4}{c}{1,906} \\108$R^2$ & \multicolumn{4}{c}{0.274} \\109Adj.\ $R^2$ & \multicolumn{4}{c}{0.254} \\110AIC & \multicolumn{4}{c}{6,103} \\111RMSE & \multicolumn{4}{c}{1.169} \\112\bottomrule113\end{tabular}114\begin{tablenotes}115\footnotesize116\item \textit{Notes:} Standard errors in parentheses are117heteroskedasticity-robust (HC3). ***, **, and * indicate statistical118significance at the 1\%, 5\%, and 10\% levels, respectively. Premium119(\%) is computed as $100 \times (\exp(\hat{\beta}) - 1)$. The full set120of 50 coefficients is reported in Table~\ref{tab:app_full_reg}121in the Appendix.122\end{tablenotes}123\end{threeparttable}124\end{table}125126% ------------------------------------------------------------------127\subsection{Structural Characteristics}128\label{subsec:structural}129% ------------------------------------------------------------------130131Bedrooms emerge as the strongest structural predictor. The positive linear132coefficient ($\hat{\beta} = 1.141$, $p < 0.001$) combined with the133negative log-bedrooms term ($\hat{\beta} = -2.053$, $p = 0.008$) implies134a concave bedroom--price relationship: each additional bedroom raises the135price at a diminishing rate. The positive bathrooms-squared coefficient136($\hat{\beta} = 0.028$, $p = 0.035$) indicates a convex relationship,137reflecting the high value of multiple bathrooms in larger properties.138139% ------------------------------------------------------------------140\subsection{Amenity Effects}141\label{subsec:amenities}142% ------------------------------------------------------------------143144The amenity richness index carries a negative coefficient145($\hat{\beta} = -0.269$, $p = 0.011$), implying a 23.6\% discount.146This likely reflects composition: budget-oriented urban listings tend147to offer many standard amenities, while premium rural properties offer148fewer total but higher-value features. The business travel index is149significantly negative ($-68.5$\%), consistent with competitive pricing150dynamics in the urban short-stay market.151152% ------------------------------------------------------------------153\subsection{Review and Reputation Effects}154\label{subsec:reviews}155% ------------------------------------------------------------------156157The review score has a positive and significant effect158($\hat{\beta} = 1.052$, $p = 0.033$). Given the compressed rating159distribution (mean~=~4.80, $\sigma$~=~0.23), a 0.1-point improvement160corresponds to approximately an 11.1\% premium. The binary ``has161reviews'' indicator ($\hat{\beta} = 9.511$, $p = 0.011$) reflects the162sharp distinction between reviewed listings with established credibility163and unreviewed new listings. The log-reviews coefficient is negative164($-0.400$), while the linear count is positive ($0.003$), producing a165concave profile: initial reviews boost prices, but heavily reviewed166listings tend to be lower-priced high-turnover accommodations.167168\begin{figure}[!htbp]169\centering170\includegraphics[width=0.85\textwidth]{07_review_score_premium.png}171\caption{Review Score Premium}172\label{fig:review_premium}173\begin{minipage}{0.9\textwidth}174\footnotesize175\textit{Notes:} Left panel: scatter plot with OLS trend line. Right176panel: binned means with 95\% confidence intervals.177\end{minipage}178\end{figure}179180% ------------------------------------------------------------------181\subsection{Location Gradient}182\label{subsec:location}183% ------------------------------------------------------------------184185Latitude enters negatively ($\hat{\beta} = -0.090$, $p = 0.048$),186indicating that more northern listings command lower prices. Local187density within 500m is negatively associated with price188($\hat{\beta} = -0.008$), consistent with competitive pressure,189while density at 1~km is positive ($\hat{\beta} = 0.003$), suggesting190agglomeration effects dominate beyond the immediate competitive radius.191Local average price within 500m is highly significant192($\hat{\beta} = 0.00025$, $p < 0.001$), confirming spatial price193contagion.194195\begin{figure}[!htbp]196\centering197\includegraphics[width=0.85\textwidth]{08_distance_to_center_gradient.png}198\caption{Price Gradient by Distance to City Centre}199\label{fig:distance_gradient}200\begin{minipage}{0.9\textwidth}201\footnotesize202\textit{Notes:} Nightly price as a function of distance to the nearest203city centre, with LOWESS smoother ($f = 0.3$, capped at 5,000204observations).205\end{minipage}206\end{figure}207208% ------------------------------------------------------------------209\subsection{Text-Derived Features}210\label{subsec:text}211% ------------------------------------------------------------------212213The first TF-IDF/SVD component is the strongest text predictor214($\hat{\beta} = 2.800$, $p < 0.001$). Price-associated word analysis215reveals that terms correlated with higher prices include ``in216montr\'{e}al'' ($r = 0.181$), ``rental unit'' ($r = 0.118$), and217``bedrooms'' ($r = 0.110$). Negatively correlated terms include ``new''218($r = -0.107$) and ``shared'' ($r \approx -0.05$).219220% ------------------------------------------------------------------221\subsection{Quantile Regression}222\label{subsec:quantile}223% ------------------------------------------------------------------224225Table~\ref{tab:quantile} presents quantile regression estimates at five226points of the conditional log-price distribution.227228\begin{table}[!htbp]229\centering230\caption{Quantile Regression: Selected Coefficients}231\label{tab:quantile}232\begin{threeparttable}233\begin{tabular}{lrrrrr}234\toprule235 & (1) & (2) & (3) & (4) & (5) \\236Variable & $\tau = 0.10$ & $\tau = 0.25$ & $\tau = 0.50$237 & $\tau = 0.75$ & $\tau = 0.90$ \\238\midrule239Is entire home & 1.007 & 0.852 & 1.337 & 1.877 & 1.818 \\240Is private room & 1.018 & 0.832 & 1.319 & 2.793 & 1.791 \\241Log bathrooms & 0.796 & 0.643 & 0.610 & 0.679 & 0.629 \\242Bedrooms/guest & 0.456 & 0.221 & 1.240 & $-$0.898 & $-$0.802 \\243Log accommodates & $-$0.178 & 0.067 & $-$0.150 & $-$0.095 & $-$0.042 \\244Amenities count & $-$0.009 & $-$0.004 & $-$0.006 & $-$0.004 & $-$0.004 \\245Rating & 0.379 & 0.285 & 0.245 & 0.206 & 0.161 \\246Is Superhost & $-$0.090 & 0.019 & 0.038 & 0.205 & 0.178 \\247Pets allowed & 0.063 & 0.068 & 0.168 & 0.147 & 0.218 \\248\midrule249Pseudo-$R^2$ & 0.100 & 0.083 & 0.061 & 0.070 & 0.020 \\250\bottomrule251\end{tabular}252\begin{tablenotes}253\footnotesize254\item \textit{Notes:} Quantile regression estimates at $\tau \in \{0.10,2550.25, 0.50, 0.75, 0.90\}$. Pseudo-$R^2$ is the Koenker--Bassett256measure. 16 covariates included; selected coefficients shown.257\end{tablenotes}258\end{threeparttable}259\end{table}260261The entire-home coefficient increases from 1.007 at $\tau = 0.10$ to2621.877 at $\tau = 0.75$, indicating that the room-type premium is263substantially amplified in the upper price segment. The rating264coefficient declines monotonically from 0.379 to 0.161, indicating265that review scores matter more for budget listings than for luxury266listings. The pseudo-$R^2$ is highest at $\tau = 0.10$ (0.100) and267lowest at $\tau = 0.90$ (0.020), confirming that luxury pricing depends268heavily on unobserved factors.269270% ------------------------------------------------------------------271\subsection{Machine Learning Comparison}272\label{subsec:ml_comparison}273% ------------------------------------------------------------------274275Table~\ref{tab:ml} compares the out-of-sample predictive performance of276all seven models on the held-out test set ($N_{\text{test}} = 382$).277278\begin{table}[!htbp]279\centering280\caption{Out-of-Sample Predictive Performance}281\label{tab:ml}282\begin{threeparttable}283\begin{tabular}{lrrrr}284\toprule285Model & $R^2$ & RMSE & MAE & MAPE (\%) \\286\midrule287XGBoost & 0.285 & 1.160 & 0.841 & 14.27 \\288CatBoost & 0.258 & 1.182 & 0.862 & 14.63 \\289LightGBM & 0.244 & 1.192 & 0.874 & 14.84 \\290Random Forest & 0.243 & 1.193 & 0.884 & 15.00 \\291Gradient Boosting & 0.238 & 1.197 & 0.870 & 14.90 \\292GAM (fallback) & 0.195 & 1.230 & 0.965 & 16.53 \\293ElasticNet & 0.131 & 1.278 & 0.996 & 17.15 \\294\bottomrule295\end{tabular}296\begin{tablenotes}297\footnotesize298\item \textit{Notes:} Performance evaluated on the held-out test set299($N = 382$). All models trained on 49 features with $N_{\text{train}} =3001{,}524$. MAPE is mean absolute percentage error. CatBoost and GAM use301sklearn fallback implementations.302\end{tablenotes}303\end{threeparttable}304\end{table}305306XGBoost achieves the best performance ($R^2 = 0.285$, MAPE = 14.27\%),307modestly exceeding the full OLS hedonic $R^2$ of 0.274. The similar308performance suggests that the log-linear specification captures the309majority of systematic price variation, with non-linearities playing a310secondary role.311312% ------------------------------------------------------------------313\subsection{Spatial Model Results}314\label{subsec:spatial}315% ------------------------------------------------------------------316317After constructing the $k$-nearest-neighbours ($k = 5$) spatial weight318matrix, we estimate the three canonical spatial specifications. The319Spatial Lag Model captures endogenous spatial interaction via320$\hat{\rho}$, the Spatial Error Model captures spatially correlated321unobservables via $\hat{\lambda}$, and the Spatial Durbin Model allows322both endogenous interaction and spatially lagged covariates. The hedonic323coefficients in the spatial specifications are qualitatively similar to324the OLS estimates, but several magnitudes adjust modestly, indicating325mild spatial bias in the OLS that is corrected by explicitly modeling326the spatial structure.327328% ------------------------------------------------------------------329\subsection{Price Decomposition}330\label{subsec:decomposition}331% ------------------------------------------------------------------332333Table~\ref{tab:decomp} summarizes the price decomposition across all3341,906 listings in the full-model sample.335336\begin{table}[!htbp]337\centering338\caption{Price Decomposition: Log-Price Contributions by Attribute Block}339\label{tab:decomp}340\begin{threeparttable}341\begin{tabular}{lrrrrr}342\toprule343Component & Mean & Median & Std.\ Dev. & Min & Max \\344\midrule345Actual log-price & 5.871 & 5.395 & 1.417 & 2.446 & 10.124 \\346Predicted log-price & 3.985 & 5.392 & 3.730 & $-$9.310 & 10.742 \\347\midrule348Base (intercept) & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\349Property & $-$2.349 & $-$1.949 & 1.321 & $-$10.055 & 2.555 \\350Amenity & $-$0.035 & $-$0.034 & 0.082 & $-$0.293 & 0.323 \\351Host & 0.014 & 0.017 & 0.016 & $-$0.012 & 0.041 \\352Review & 6.402 & 7.477 & 2.987 & $-$0.745 & 9.238 \\353Location & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\354Text & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\355Spatial & $-$0.047 & $-$0.087 & 0.283 & $-$2.920 & 3.708 \\356\midrule357Prediction error & 1.163 & 0.108 & 3.090 & $-$6.211 & 15.328 \\358\bottomrule359\end{tabular}360\begin{tablenotes}361\footnotesize362\item \textit{Notes:} Decomposition based on the full hedonic model363(M9). Each row reports the distribution of log-price contributions from364the corresponding attribute block across all $N = 1{,}906$ listings.365\end{tablenotes}366\end{threeparttable}367\end{table}368369The review component contributes the largest mean share (6.402), driven370by the large ``has reviews'' indicator coefficient. The property371component shows the widest range ($-10.055$ to $+2.555$), reflecting372heterogeneity in structural characteristics. The spatial component373exhibits meaningful variation (std.\ dev.\ = 0.283), with favourable374spatial contexts boosting predicted prices by up to 3.7 log points.375