% Author: Simon-Pierre Boucher — contact@spboucher.ai
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% 05_results.tex
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\section{Results}\label{sec:results}

This section presents the empirical results for each of the six models described in Section~\ref{sec:methodology}.  Throughout, we use cautious language to reflect the conditional nature of our estimates: the terms ``associated with'' and ``conditional correlation'' are used in place of ``effect'' or ``impact,'' consistent with the limitations of our cross-sectional identification strategy.

\subsection{Baseline Hedonic Rent Model}

Table~\ref{tab:hedonic_rent} reports the OLS estimates of the hedonic rent model (Equation~\ref{eq:hedonic_rent}).  Column~(1a) presents the bivariate specification with only the Airbnb count within 500\,m; column~(1b) adds dwelling controls (bedrooms, bathrooms, building type); and column~(1c)---our preferred specification---adds city fixed effects.  Columns~(1d) and~(1e) replace the Airbnb count with two alternative exposure measures, the Airbnb density and the share of entire-home listings within the buffer.

\begin{table}[htbp]
    \centering
    \caption{Hedonic Rent Model --- Baseline OLS Results}
    \label{tab:hedonic_rent}
    \input{../results/tables/hedonic_rent_baseline.tex}
\end{table}

The coefficient on $\texttt{airbnb\_count\_500m}$ is positive and statistically significant across all specifications.  In the preferred specification (column~1c), each additional Airbnb listing within 500\,m is associated with an approximate 0.4\% increase in monthly rent, holding dwelling characteristics and city constant.  This semi-elasticity is economically modest but statistically robust: at the median monthly rent of approximately \$1{,}950, one additional nearby Airbnb listing corresponds to a rent differential of roughly \$6--\$10 per month.  However, the practical significance compounds when one considers that many urban rental listings have 10 or more Airbnb listings within 500\,m, implying cumulative differentials that are economically meaningful.

Among the control variables, dwelling size is the dominant predictor of rent: each additional bedroom is associated with approximately 11--13\% higher rent, and each additional bathroom with approximately 23--30\% higher rent.  Building type (House and Row/Townhouse relative to Apartment) also enters significantly.  City fixed effects raise the explanatory power of the model considerably, confirming substantial cross-city variation in rent levels.

The adjusted $R^2$ of the preferred specification is approximately 0.56, indicating that observed dwelling characteristics and city fixed effects explain a substantial share of the cross-sectional variation in rents, though a considerable residual remains, consistent with the importance of unobserved unit-specific and micro-locational factors.

\subsection{Hedonic Airbnb Pricing Model}

Table~\ref{tab:hedonic_airbnb} reports the estimates of the Airbnb pricing model (Equation~\ref{eq:hedonic_airbnb}).

\begin{table}[htbp]
    \centering
    \caption{Hedonic Airbnb Pricing Model --- OLS Results}
    \label{tab:hedonic_airbnb}
    \input{../results/tables/hedonic_airbnb_pricing.tex}
\end{table}

Contrary to the opportunity-cost hypothesis, the city-level mean rent does not enter significantly in any specification: the coefficient $\hat{\theta}$ is small and statistically indistinguishable from zero, with a negative point estimate in columns~(2a) and~(2b) and a positive one in column~(2c).  Airbnb nightly prices in our sample are therefore not systematically higher in higher-rent cities once listing characteristics are taken into account; the pricing of short-term rentals appears to be driven primarily by the properties of the listing itself rather than by conditions in the local long-term rental market.

Among the listing-level controls, dwelling size matters most: bedrooms and, especially, bathrooms are associated with significantly higher nightly prices, in line with the STR pricing literature \citep{wang2017price, gibbs2018pricing}.  Superhost status carries a significant \textit{negative} coefficient---a notable departure from the positive reputation premia typically reported \citep{wang2017price, ert2016trust}---which likely reflects composition effects: superhosts in our Quebec sample are concentrated in more modest, high-volume urban units rather than in the luxury chalet segment, and guest capacity enters negatively once size is controlled for.  The entire-home indicator is positive and significant once city fixed effects are included (column~2c), consistent with whole units commanding a premium over rooms in shared dwellings within the same market \citep{gibbs2018pricing}.

When city fixed effects are included, the listing-level coefficients retain their signs and broad magnitudes, suggesting that the within-city pricing structure of Airbnb listings is largely independent of the cross-city rent variation.

\subsection{City-Level Results}

Table~\ref{tab:city_level} presents the city-level regressions (Equations~\ref{eq:city_rent} and~\ref{eq:city_airbnb}).

\begin{table}[htbp]
    \centering
    \caption{City-Level Regressions}
    \label{tab:city_level}
    \input{../results/tables/city_level_interaction.tex}
\end{table}

The forward regression shows a positive and statistically significant---though economically small---cross-city association between the total number of Airbnb listings and mean log rents: cities with more Airbnb listings tend to have somewhat higher average rents, conditional on average dwelling size.  The reverse regression, by contrast, has essentially no explanatory power ($R^2 = 0.01$), and mean rent does not significantly predict city-level Airbnb counts.  Although the aggregation yields 153 city-level observations, most cities contribute only a handful of underlying listings, which limits the statistical power of these regressions and amplifies the influence of individual city outliers.  We interpret these results as descriptive patterns that motivate the listing-level analysis rather than as evidence of a causal relationship.

\subsection{Spatial Model Results}

Table~\ref{tab:spatial_results} reports the OLS baseline alongside the spatial lag (SAR) and spatial error (SEM) models of Equation~\ref{eq:spatial_lag}.

\begin{table}[htbp]
    \centering
    \caption{Spatial Regression Models}
    \label{tab:spatial_results}
    \input{../results/tables/spatial_models.tex}
\end{table}

The spatial autoregressive coefficient ($\hat{\rho}$, the coefficient on $W \cdot \ln(\text{rent})$) is positive and strongly significant, and the spatial error parameter $\hat{\lambda}$ in the SEM is large (0.54), confirming substantial spatial dependence in rents: the rents of nearby listings carry systematic information about a given listing's rent, even after controlling for dwelling characteristics and city fixed effects.  This finding is consistent with the large literature on spatial autocorrelation in housing markets.

Importantly, the coefficient on $\texttt{airbnb\_count\_500m}$ ($\hat{\beta}'$) remains positive and statistically significant in both spatial specifications: it is mildly attenuated in the SAR model (0.0034 versus 0.0038 in the OLS baseline) and essentially unchanged in the SEM.  The mild attenuation is expected: to the extent that the Airbnb coefficient in the baseline model partly reflected spatially correlated omitted variables (captured by the spatial terms), the spatial models provide more conservative estimates.  The persistence of a significant positive association strengthens confidence that the Airbnb--rent correlation is not solely an artefact of spatial confounding.

Figure~\ref{fig:buffer_decay} illustrates the spatial decay of the Airbnb coefficient across buffer radii.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.75\textwidth]{coefficient_buffer_comparison.pdf}
    \caption{Estimated Airbnb Coefficient by Buffer Radius}
    \label{fig:buffer_decay}
\end{figure}

The per-listing coefficient is largest at the 250\,m radius and declines monotonically as the buffer expands, consistent with a localised association between Airbnb activity and rents.  At the 2\,km radius, the coefficient remains positive but is substantially smaller in magnitude, reflecting the dilution of the Airbnb signal over a larger geographic area.  This spatial decay pattern is consistent with prior findings in the literature and suggests that the Airbnb--rent association operates at a highly localised scale.

\subsection{Quantile Regression Results}

Table~\ref{tab:quantile} and Figure~\ref{fig:quantile_plot} present the quantile regression estimates of $\beta_\tau$ for $\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$.

\begin{table}[htbp]
    \centering
    \caption{Quantile Regression Results --- Coefficient on Airbnb Count (500m)}
    \label{tab:quantile}
    \small\setlength{\tabcolsep}{4pt}
    \input{../results/tables/quantile_regression.tex}
\end{table}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.75\textwidth]{quantile_coefficients.pdf}
    \caption{Quantile Regression Coefficients on Airbnb Count (500m) with 95\% Confidence Intervals}
    \label{fig:quantile_plot}
\end{figure}

The quantile regression results reveal meaningful heterogeneity in the Airbnb--rent association across the conditional rent distribution.  The coefficient is positive and statistically significant at every quantile considered.  It is roughly flat over the lower half of the distribution (0.0037 at $\tau = 0.10$, 0.0034--0.0035 at $\tau = 0.25$ and $\tau = 0.50$) and then rises in the upper tail, reaching 0.0041 at $\tau = 0.75$ and its largest value, 0.0047, at $\tau = 0.90$---roughly 25\% above the OLS estimate.

This pattern is economically interpretable.  High-rent listings---typically located in desirable neighbourhoods with strong tourist appeal---are precisely the locations where Airbnb activity is most intense and where the supply-withdrawal mechanism is most plausible.  The finding that implicit prices vary across the conditional distribution echoes the broader hedonic evidence \citep{zietz2008determinants, mcmillen2008changes}, and extends it to STR exposure: the association between Airbnb and rents, while present throughout the distribution, is strongest in the upper segment of the rental market.

\subsection{Machine-Learning Robustness}

Table~\ref{tab:ml_performance} reports the out-of-sample predictive performance of the four machine-learning models alongside the OLS benchmark.

\begin{table}[htbp]
    \centering
    \caption{Machine-Learning Model Performance (Test Set)}
    \label{tab:ml_performance}
    \input{../results/tables/ml_comparison.tex}
\end{table}

The random forest achieves the highest $R^2$ on the test set (0.71), closely followed by gradient boosting (0.69); the regularised linear models (LASSO and elastic net) perform essentially on par with OLS (test $R^2 \approx 0.51$).  The gap between the tree-based and linear models indicates that nonlinear relationships and interactions among covariates---most plausibly involving the geographic coordinates---contribute meaningfully to rent variation beyond what the linear hedonic model captures.  The linear specification nevertheless accounts for the bulk of the explainable variation and remains a reasonable approximation for inference on the Airbnb exposure coefficient.

Figure~\ref{fig:feature_importance} displays the mean absolute SHAP values from the gradient boosting model.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.80\textwidth]{feature_importance.pdf}
    \caption{Feature Importance from Gradient Boosting Model}
    \label{fig:feature_importance}
\end{figure}

Dwelling characteristics (bathrooms, bedrooms) and geographic coordinates are consistently the most important predictors of rent.  The Airbnb exposure variables---mean nearby price, density, count, and entire-home share---each rank as moderately important features and jointly account for a non-trivial share of predictive power, confirming that Airbnb exposure carries information beyond what is captured by dwelling size and raw location.  The LASSO and elastic net models retain the Airbnb count variable with a positive coefficient, consistent with the OLS results.

Figure~\ref{fig:shap_summary} presents a SHAP summary plot, providing a more granular view of how each feature contributes to individual predictions.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.80\textwidth]{shap_summary.pdf}
    \caption{SHAP Summary Plot from Gradient Boosting Model}
    \label{fig:shap_summary}
\end{figure}

The SHAP analysis confirms that higher Airbnb counts are associated with positive contributions to predicted rent, consistent with the parametric results.  The distribution of SHAP values for the Airbnb count variable shows a rightward shift for listings with high Airbnb exposure, reinforcing the finding that nearby short-term rental activity is associated with higher rents.
