spb/wp4_uqo

1% ==========================================================================2\section{Robustness}3\label{sec:robustness}4% ==========================================================================56Table~\ref{tab:robustness} re-estimates the M5 specification under7alternative definitions of the sample and of the fixed effects and8reports the coefficients that carry the paper's main results. The9checks address three threats: that the results depend on how the10stay-price regime was removed, that they are driven by a particular11segment of the market, and that they depend on the spatial resolution12of the fixed effects.1314\begin{table}[!htbp]15\centering16\caption{Robustness of the Main Coefficients (Specification M5)}17\label{tab:robustness}18\begin{threeparttable}19\scriptsize20\setlength{\tabcolsep}{3pt}21\input{../results/tables/rev_robustness.tex}22\begin{tablenotes}23\footnotesize24\item \textit{Notes:} Each row re-estimates M5 (all regressors and city /25cell fixed effects) on the indicated sample; HC3 standard errors in26parentheses. ``Bedr.'', ``Bathr.'' and ``Guests'' are the elasticities27with respect to bedrooms, one plus bathrooms and guest capacity;28``Rating'' is the coefficient on the centred rating among rated29listings; ``ln dist.'' is the coefficient on $\ln(1+\text{km to30centre})$. $\hat p_i$ is the posterior probability of the stay regime31from Equation~\eqref{eq:mixture}. The ``weighted'' row uses weighted32least squares with weights $1 - \hat p_i$ on the baseline sample. The33last row replaces the city / cell fixed effects by 60 $k$-means spatial34cells. ***, ** and * denote significance at the 1, 5 and 10\% levels.35\end{tablenotes}36\end{threeparttable}37\end{table}3839\paragraph{The regime classification.} The baseline results are40insensitive to the posterior cut-off: tightening it to 0.25 removes five41listings and loosening it to 0.75 adds two, and no coefficient moves by42more than 0.02. Weighting observations by $1 - \hat p_i$ instead of43classifying them gives the same estimates. The comparison with the44simple price rules is instructive. Keeping every listing priced at or45below \$1,000, the rule used in the companion paper, leaves 8346stay-regime listings in the sample; because these are small, cheap-per-night47units, they pull the capacity elasticity to zero (0.015 against 0.331),48halve the rating coefficient and lower $R^2$ to 0.28. A \$600 cut-off49removes all but nine of them but also discards 87 genuine nightly prices50of large chalets and houses, which compresses the bathroom and rating51coefficients (0.247 and 0.254) and lowers $R^2$ to 0.42. The full priced52sample, finally, reproduces the pathology of the first version of this53paper: the capacity elasticity turns negative, the badges and the review54count acquire large negative coefficients, the rating coefficient55doubles, and $R^2$ is 0.16. The model-based classification is therefore56not a cosmetic choice; the price rules are biased in opposite directions57and the bias is concentrated in exactly the coefficients of interest.5859\paragraph{Composition.} Excluding hotel rooms, bed-and-breakfasts and60hostels, which are priced per room on a different logic, raises the61bathroom elasticity to 0.54 and the rating coefficient to 0.38 and62leaves the rest unchanged. Restricting the sample to rated listings, so63that the rating enters for every observation, gives the same rating64coefficient (0.37) and confirms that the unrated indicator is not driving65it. Splitting the sample at Montreal reveals the one substantive66heterogeneity in the table. Within Montreal ($N = 506$) the bedroom67elasticity is small and imprecise, the review count is positively priced68(0.112, standard error 0.021) and the distance gradient is steep69($-0.173$ per log-kilometre); outside Montreal the bedroom elasticity is700.22, reviews are unpriced and distance to the nearest centre does not71matter. In the metropolitan market, where listings are substitutes72within a dense grid, an established review record and proximity to the73centre are what differentiate listings; in the resort and small-town74markets, size does. The capacity, bathroom and rating coefficients are75nonetheless similar in the two sub-samples, and the within-Montreal76$R^2$ (0.54) is slightly higher than the provincial one.7778\paragraph{Outliers and fixed effects.} Trimming the top and bottom79percentile of prices within the analysis sample lowers the bathroom80elasticity to 0.30 and leaves the other coefficients unchanged; the81largest bathroom counts belong to the most expensive chalets, so the82bathroom coefficient is partly identified from the upper tail, as the83quantile results already indicated. Replacing the 51 city and cell fixed84effects by 60 finer spatial cells, which halves the average number of85listings per group, changes no coefficient by more than 0.03 and leaves86$R^2$ at 0.505, so the listing-level implicit prices are not sensitive87to how finely location is controlled. The spatial models of88Section~\ref{subsec:spatial} provide the complementary check that the89coefficients are robust to modelling the dependence explicitly rather90than absorbing it: with ten rather than five nearest neighbours the91spatial lag coefficient is 0.215 (standard error 0.027) and the Moran92statistic of the M4 residuals is 0.094 ($z = 11.4$), so the dependence is93not an artefact of a particular neighbourhood definition.9495\medskip96In summary, the implicit prices of capacity, bathrooms, the shared97bathroom, photos, the rating and local density, and the absence of a98Superhost or Guest Favourite premium, hold across the regime cut-off,99weighting, the exclusion of commercial lodging, trimming, finer fixed100effects and alternative spatial weights. The two results that are101sample-dependent are the bedroom elasticity, which is driven by the102non-metropolitan market, and the review-count coefficient, which is103positive in Montreal and zero elsewhere; both are consistent with the104quantile patterns and are reported as heterogeneity rather than as105instability.106