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\section{Robustness}
\label{sec:robustness}
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Table~\ref{tab:robustness} re-estimates the M5 specification under
alternative definitions of the sample and of the fixed effects and
reports the coefficients that carry the paper's main results. The
checks address three threats: that the results depend on how the
stay-price regime was removed, that they are driven by a particular
segment of the market, and that they depend on the spatial resolution
of the fixed effects.

\begin{table}[!htbp]
\centering
\caption{Robustness of the Main Coefficients (Specification M5)}
\label{tab:robustness}
\begin{threeparttable}
\scriptsize
\setlength{\tabcolsep}{3pt}
\input{../results/tables/rev_robustness.tex}
\begin{tablenotes}
\footnotesize
\item \textit{Notes:} Each row re-estimates M5 (all regressors and city /
cell fixed effects) on the indicated sample; HC3 standard errors in
parentheses. ``Bedr.'', ``Bathr.'' and ``Guests'' are the elasticities
with respect to bedrooms, one plus bathrooms and guest capacity;
``Rating'' is the coefficient on the centred rating among rated
listings; ``ln dist.'' is the coefficient on $\ln(1+\text{km to
centre})$. $\hat p_i$ is the posterior probability of the stay regime
from Equation~\eqref{eq:mixture}. The ``weighted'' row uses weighted
least squares with weights $1 - \hat p_i$ on the baseline sample. The
last row replaces the city / cell fixed effects by 60 $k$-means spatial
cells. ***, ** and * denote significance at the 1, 5 and 10\% levels.
\end{tablenotes}
\end{threeparttable}
\end{table}

\paragraph{The regime classification.} The baseline results are
insensitive to the posterior cut-off: tightening it to 0.25 removes five
listings and loosening it to 0.75 adds two, and no coefficient moves by
more than 0.02. Weighting observations by $1 - \hat p_i$ instead of
classifying them gives the same estimates. The comparison with the
simple price rules is instructive. Keeping every listing priced at or
below \$1,000, the rule used in the companion paper, leaves 83
stay-regime listings in the sample; because these are small, cheap-per-night
units, they pull the capacity elasticity to zero (0.015 against 0.331),
halve the rating coefficient and lower $R^2$ to 0.28. A \$600 cut-off
removes all but nine of them but also discards 87 genuine nightly prices
of large chalets and houses, which compresses the bathroom and rating
coefficients (0.247 and 0.254) and lowers $R^2$ to 0.42. The full priced
sample, finally, reproduces the pathology of the first version of this
paper: the capacity elasticity turns negative, the badges and the review
count acquire large negative coefficients, the rating coefficient
doubles, and $R^2$ is 0.16. The model-based classification is therefore
not a cosmetic choice; the price rules are biased in opposite directions
and the bias is concentrated in exactly the coefficients of interest.

\paragraph{Composition.} Excluding hotel rooms, bed-and-breakfasts and
hostels, which are priced per room on a different logic, raises the
bathroom elasticity to 0.54 and the rating coefficient to 0.38 and
leaves the rest unchanged. Restricting the sample to rated listings, so
that the rating enters for every observation, gives the same rating
coefficient (0.37) and confirms that the unrated indicator is not driving
it. Splitting the sample at Montreal reveals the one substantive
heterogeneity in the table. Within Montreal ($N = 506$) the bedroom
elasticity is small and imprecise, the review count is positively priced
(0.112, standard error 0.021) and the distance gradient is steep
($-0.173$ per log-kilometre); outside Montreal the bedroom elasticity is
0.22, reviews are unpriced and distance to the nearest centre does not
matter. In the metropolitan market, where listings are substitutes
within a dense grid, an established review record and proximity to the
centre are what differentiate listings; in the resort and small-town
markets, size does. The capacity, bathroom and rating coefficients are
nonetheless similar in the two sub-samples, and the within-Montreal
$R^2$ (0.54) is slightly higher than the provincial one.

\paragraph{Outliers and fixed effects.} Trimming the top and bottom
percentile of prices within the analysis sample lowers the bathroom
elasticity to 0.30 and leaves the other coefficients unchanged; the
largest bathroom counts belong to the most expensive chalets, so the
bathroom coefficient is partly identified from the upper tail, as the
quantile results already indicated. Replacing the 51 city and cell fixed
effects by 60 finer spatial cells, which halves the average number of
listings per group, changes no coefficient by more than 0.03 and leaves
$R^2$ at 0.505, so the listing-level implicit prices are not sensitive
to how finely location is controlled. The spatial models of
Section~\ref{subsec:spatial} provide the complementary check that the
coefficients are robust to modelling the dependence explicitly rather
than absorbing it: with ten rather than five nearest neighbours the
spatial lag coefficient is 0.215 (standard error 0.027) and the Moran
statistic of the M4 residuals is 0.094 ($z = 11.4$), so the dependence is
not an artefact of a particular neighbourhood definition.

\medskip
In summary, the implicit prices of capacity, bathrooms, the shared
bathroom, photos, the rating and local density, and the absence of a
Superhost or Guest Favourite premium, hold across the regime cut-off,
weighting, the exclusion of commercial lodging, trimming, finer fixed
effects and alternative spatial weights. The two results that are
sample-dependent are the bedroom elasticity, which is driven by the
non-metropolitan market, and the review-count coefficient, which is
positive in Montreal and zero elsewhere; both are consistent with the
quantile patterns and are reported as heterogeneity rather than as
instability.