spb/wp1_uqo

1% ============================================================================2\section{Robustness and the limits of the available evidence}3\label{sec:robustness}4% ============================================================================56A revision of this paper carried out in October 2026 without access to the Centris snapshot (Appendix~\ref{app:audit}) made it necessary to distinguish sharply between robustness evidence that exists in the numerical record and robustness checks that were described in the May 2026 version but for which no output survives. This section reports the former, states the latter as not executed, and ranks the checks that a re-run of the pipeline should perform first.78\subsection{Evidence in the record}910Three internal checks are available. First, the heterogeneity results for houses do not depend on a single pair of quantiles: the ranking of the amenities by the equality statistic is the same for the $(0.10, 0.90)$, $(0.25, 0.75)$ and $(0.10, 0.50)$ pairs (Table~\ref{tab:wald_houses}), and the adjacent-pair statistics of Figure~\ref{fig:wald} locate the changes in the lower half of the distribution rather than at a single extreme quantile, which is where estimates would be most sensitive to trimming. Second, the direction of the quantile profiles is consistent with the direction of the OLS-to-2SQR attenuation: the indicators whose coefficients decline most across quantiles (highway, transit) are also those most attenuated by the spatial lag at the median (Table~\ref{tab:sqr}), which is what one expects if both phenomena reflect the clustering of lower-priced, amenity-dense listings. Third, the cross-segment comparison is itself a specification check: the transit and school differentials have the same sign in the three built segments, and the segment in which precision is lowest (condominiums) is the one in which no heterogeneity is detected, whereas the two largest samples (houses, land) produce the sharpest rejections. None of these observations substitutes for the formal checks below.1112\subsection{Checks described in the previous version but not supported by the record}1314The May 2026 version reported a sensitivity analysis of $\hat\rho$ and of the amenity coefficients to the number of neighbours $k \in \{5, 10, 15, 20\}$, a comparison of bootstrap standard errors at $B = 200$ and $B = 500$, re-estimation under 1/99 and 2/98 percent trimming, a split of the house sample into Montreal and non-Montreal listings, variance inflation factors and a condition number for the house design matrix \citep{belsley1980regression, obrien2007caution}, a partial first-stage $F$ above 100 in every segment, and a statement that the scalar effect decomposition was within 3 percent of the trace-based LeSage--Pace effects on a subsample. None of these outputs is in the preserved log, the figures reported for them could not be reconciled with the log, and the current version of the estimation script computes only the variance inflation factors, the condition number and the overall first-stage $F$. We therefore withdraw these claims. They are listed in Appendix~\ref{app:audit} as analyses to be executed, and the paper's conclusions are stated so as not to depend on them.1516\subsection{What a re-run should establish first}17\label{sec:limitations}1819The checks below are ordered by their bearing on the conclusions. The first concerns inference. The paired bootstrap ignores spatial dependence that is, by the paper's own diagnostics, strong; a cluster bootstrap by municipality or by spatial block \citep{cameron2008bootstrap, parente2016quantile, hagemann2017cluster} would widen the confidence bands, and the equality tests of Table~\ref{tab:wald} should be recomputed with the bootstrap covariance between quantiles, which can only make them less conservative, and with the full stacked covariance matrix to obtain a valid joint test \citep{koenker2002inference}. The second concerns the 2SQR itself: the second stage should be bootstrapped jointly with the first to obtain standard errors for $\hat\rho(\tau)$ and for the spatially adjusted amenity coefficients, the partial $F$ on $\mathbf{WZ}$ should be reported, the structural residuals should be used for Moran's $I$, and the quantile-based first stage of \citet{kim2004quantile} or the inverse quantile regression of \citet{chernozhukov2006quantile} and \citet{su2011instrumental} should be compared with the least-squares first stage. The third concerns the spatial weights: $k$, projected rather than geographic coordinates, a positive distance floor for co-located listings and a distance-band alternative. The fourth concerns the specification: the scaling of the distance variable (Section~\ref{sec:results}), the replacement of a single province-wide distance gradient by municipality fixed effects (for which the location-scale quantile estimator of \citet{machado2019quantiles} handles many fixed effects at low cost), and the sensitivity of the tails to the 0.5 percent trimming. The fifth concerns the data: the broker's amenity classification should be validated against geocoded distances to actual facilities for a subsample, and the asking-price results replicated on transaction prices from the Quebec land register (\emph{Registre foncier}). Until at least the first two groups of checks have been run, the heterogeneity results should be regarded as suggestive rather than established, and the spatial-lag estimates as descriptive.20