% Author: Simon-Pierre Boucher — contact@spboucher.ai
% =============================================================================
% 06_robustness.tex
% =============================================================================
\section{Robustness Checks}\label{sec:robustness}

This section presents a battery of robustness tests designed to assess the sensitivity of our baseline estimates to alternative specifications, sample definitions, and exposure measures.

\subsection{Buffer Radius Robustness}

Our baseline specification uses a 500\,m buffer to define Airbnb exposure.  Table~\ref{tab:buffer_robustness} reports the coefficient on the Airbnb count variable for buffer radii of 250\,m, 500\,m, 1\,km, and 2\,km, estimated using the same hedonic specification (Equation~\ref{eq:hedonic_rent}) with the full set of dwelling controls and city fixed effects.

\begin{table}[htbp]
    \centering
    \caption{Buffer Radius Robustness --- Hedonic Rent Model}
    \label{tab:buffer_robustness}
    \input{../results/tables/buffer_robustness.tex}
\end{table}

The results confirm the spatial decay pattern illustrated in Figure~\ref{fig:buffer_decay}.  The per-listing coefficient is largest at the 250\,m radius, where the spatial signal is most concentrated, and decreases monotonically as the buffer expands---mechanically so, since a listing counted within a wide buffer is a weaker proxy for immediate proximity than one counted within a narrow buffer.  At all radii, the coefficient is positive and statistically significant at the 1\% level.  The consistency of the positive association across radii provides reassurance that the finding is not an artefact of the particular buffer choice.

Figure~\ref{fig:buffer_coef_plot} visualises the coefficient estimates and their 95\% confidence intervals across buffer radii.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.70\textwidth]{coefficient_buffer_comparison.pdf}
    \caption{Airbnb Count Coefficient Across Buffer Radii (with 95\% CI)}
    \label{fig:buffer_coef_plot}
\end{figure}

\subsection{Alternative Exposure Measures}

Our baseline uses the raw count of Airbnb listings as the exposure measure.  We consider two alternative measures to assess whether the results are sensitive to the functional form of the exposure variable:

\begin{enumerate}[label=(\roman*)]
    \item \textbf{Airbnb density:} the count divided by the buffer area ($\pi r^2$ in km$^2$), which normalises for the geometric expansion of the buffer.
    \item \textbf{Share of entire-home listings:} the fraction of Airbnb listings within the buffer that are classified as entire-home properties (House, Cabin/Chalet, Condo), capturing the composition of short-term rental activity.
\end{enumerate}

These alternative specifications are reported as Models~(1d) and~(1e) in Table~\ref{tab:hedonic_rent}.

The Airbnb density measure yields qualitatively identical results to the raw count, with a positive and significant coefficient---as expected, since at a fixed radius the density is a rescaling of the count.  The share of entire-home listings, by contrast, is \textit{not} significantly associated with rents (the point estimate is small and negative): conditional on dwelling characteristics and city fixed effects, it is the intensity of nearby Airbnb activity, rather than its compositional tilt toward entire homes, that co-varies with rents in our data.  This finding does not support the compositional prediction of the supply-withdrawal hypothesis, under which entire-home listings---the closest substitutes for long-term rental units---should matter most; we return to this point in Section~\ref{sec:discussion}.

\subsection{Subsample Analysis}

To assess whether the baseline results are driven by a particular geographic segment or building type, we estimate the hedonic rent model separately for the following subsamples:

\begin{enumerate}[label=(\roman*)]
    \item \textbf{Montreal vs.\ non-Montreal:} Montreal dominates both datasets and is the primary tourist destination.  The Airbnb--rent association may be stronger in Montreal, where short-term rental activity is most concentrated, or weaker if the city's larger and more liquid rental market is better able to absorb the supply shock.
    \item \textbf{Apartments vs.\ Houses:} Apartments constitute the vast majority of rental listings and are arguably the closest substitute for Airbnb rental units.  Houses, by contrast, are a more heterogeneous category and may operate in a partially segmented market.
\end{enumerate}

Table~\ref{tab:subsample} presents the results.

\begin{table}[htbp]
    \centering
    \caption{Subsample Analysis --- Hedonic Rent Model}
    \label{tab:subsample}
    \input{../results/tables/robustness_subsamples.tex}
\end{table}

The Montreal subsample yields a positive and significant Airbnb coefficient that is similar in magnitude to the full-sample estimate, confirming that the baseline results are not driven solely by the inclusion of non-Montreal observations.  The non-Montreal subsample also produces a positive coefficient, though with lower precision due to the smaller sample size and the greater heterogeneity of non-Montreal markets (which include both urban centres and resort communities).

The apartment subsample closely mirrors the full-sample results, which is unsurprising given that apartments constitute approximately 90\% of the rental sample.  The house subsample yields a coefficient that is larger than the full-sample estimate (0.0067) and statistically significant despite the much smaller sample, though its wider confidence interval reflects the greater heterogeneity of house rentals.

\subsection{Outlier Sensitivity}

To assess the sensitivity of the results to extreme values, we re-estimate the baseline model on two trimmed samples: one that retains only listings with monthly rents between the 5th and 95th percentiles, and one that drops the top and bottom 1\% of observations by Airbnb count within 500\,m (i.e., listings in the most Airbnb-saturated locations).

\begin{table}[htbp]
    \centering
    \caption{Outlier Sensitivity --- Hedonic Rent Model}
    \label{tab:outlier_sensitivity}
    \input{../results/tables/robustness_outliers.tex}
\end{table}

The results are stable across the outlier-handling approaches.  Trimming the rent distribution at the 5th and 95th percentiles reduces the point estimate by roughly a fifth (from 0.0039 to 0.0031)---consistent with the quantile-regression finding that the association is strongest in the tails---while trimming extreme Airbnb counts slightly increases it (0.0042).  In every case the coefficient remains positive and significant at the 1\% level, providing confidence that the baseline results are not driven by a small number of influential observations.

\subsection{Additional Specification Checks}

Table~\ref{tab:extended_robustness} subjects the preferred specification to four further checks.  Column~(R1) reproduces the baseline for reference.

\begin{table}[htbp]
    \centering
    \caption{Extended Robustness --- Preferred Hedonic Specification}
    \label{tab:extended_robustness}
    \small
    \input{../results/tables/extended_robustness.tex}
\end{table}

\paragraph{Clustered standard errors.}  Column~(R2) re-computes the baseline standard errors clustering by city (153 clusters).  HC1 standard errors assume independence across observations, which is questionable when listings are spatially concentrated.  In our data, city-level clustering in fact \textit{tightens} the standard error on the Airbnb coefficient (0.0001 versus 0.0002 under HC1), so the baseline significance levels are, if anything, conservative on this dimension.  Clustering does widen the confidence interval on some controls (bathrooms), as expected.

\paragraph{Interior size.}  Column~(R3) adds the interior floor area (per 100 sq ft) as a control on the subsample of 3{,}531 listings that report it.  Size enters positively and significantly, and the Airbnb coefficient remains positive and highly significant at 0.0032---about a fifth smaller than the baseline, consistent with size absorbing some quality variation, but leaving the qualitative conclusion intact.

\paragraph{Functional form.}  Column~(R4) replaces the linear count with $\ln(1 + \texttt{airbnb\_count\_500m})$.  The elasticity-style coefficient (0.048) is positive and highly significant: doubling the nearby Airbnb count is associated with roughly 3.4\% higher rent.  The fit improves marginally over the linear form, indicating mild concavity in the exposure--rent relationship.

\paragraph{Ring exposure.}  Column~(R5) includes the 500\,m count jointly with the count in the 500\,m--1\,km annulus.  Both enter positively and significantly.  Splitting the exposure this way attenuates the inner-buffer coefficient (0.0013), and the annulus coefficient (0.0018) is of comparable magnitude per listing: the association is therefore not confined to the immediate 500\,m but extends to the kilometre scale, consistent with the buffer-radius results above, where the total (not per-listing) association accumulates over wider areas.  Because the two counts are strongly correlated, the individual coefficients in this specification should be read as a decomposition of a common neighbourhood signal rather than as separate causal gradients.

\subsection{Leave-One-City-Out Sensitivity}

Figure~\ref{fig:leave_one_out} re-estimates the preferred specification excluding, in turn, each of the eight cities with the most rental listings.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.80\textwidth]{leave_one_city_out.pdf}
    \caption{Leave-One-City-Out Sensitivity of the Airbnb Coefficient}
    \label{fig:leave_one_out}
\end{figure}

The coefficient is remarkably stable, ranging from 0.0038 to 0.0040 across exclusions.  Notably, excluding Montreal---which accounts for more than half of the estimation sample---leaves the point estimate essentially unchanged (0.0038), albeit with a wider confidence interval; the association is thus not an artefact of any single market.

\subsection{Summary of Robustness}

Figure~\ref{fig:robustness_summary} presents a coefficient plot summarising the Airbnb count coefficient across all robustness specifications.

\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.80\textwidth]{coefficient_robustness.pdf}
    \caption{Summary of Airbnb Count Coefficients Across Specifications}
    \label{fig:robustness_summary}
\end{figure}

Across all specifications---varying the buffer radius, the exposure measure, the functional form, the control set, the sample, the outlier treatment, and the standard-error computation---the estimated association between Airbnb presence and residential rents is consistently positive.  While the magnitude varies across specifications, the qualitative conclusion is robust: higher Airbnb density in the immediate vicinity of a rental listing is associated with higher monthly rents, conditional on observable dwelling characteristics and city-level heterogeneity.
