spb/wp16_uqo

1% ============================================================================2\section{Robustness}3\label{sec:robustness}4% ============================================================================56\subsection{Stability over time}78Table~\ref{tab:stability} reports the MdAPE and bias of the main models by quarter of the test period. The rankings are stable quarter by quarter, and the validation-period (2024) and test-period (2025--2026) MdAPEs of every configuration in Tables~\ref{tab:adjustments}--\ref{tab:competition} differ by about a point in the same direction, which indicates that the rules selected on 2024 did not overfit it. The comparable-based methods display a seasonal bias: estimates are too low in the first quarter (\biasQoneMseven{} for M7) and too high in the third (\biasQthreeMseven{}). This is the signature of the point-in-time constraint---every method works at the level of the previous month, and Quebec prices rise into the spring and soften after the summer---and of the fact that the price level of the valuation month is not knowable on the valuation date. The gradient-boosting benchmark, which includes the month of sale as a feature, absorbs most of this seasonality; a seasonal adjustment estimated on prior years would do the same for the comparables method.910\begin{table}[H]11\centering12\caption{Stability over the test period: MdAPE and bias (\%) by quarter}13\label{tab:stability}14\footnotesize15\input{tables/stability}16\end{table}1718\subsection{Coverage and the common-support comparison}1920Configurations with narrow radii or short windows abstain on a substantial share of targets, and their MdAPE on the targets they cover is not comparable to that of configurations that cover all targets. Table~\ref{tab:surface_cov} in the Appendix reports the coverage of every point of the error surface. Our configuration selection enforces 97\% coverage; relaxing the constraint selects a narrower configuration (\mZeroUncDesc) with a lower MdAPE on its own support (\mZeroUncMdape{}) but \mZeroUncCov{} coverage. The conclusions of Section~\ref{sec:surface} hold on the common support of all configurations with at least 97\% coverage.2122\subsection{Exclusion rules}2324The exclusion of extreme price-to-assessment ratios (rule R8) is the only rule that removes a non-trivial share of sales on a criterion related to price. The excluded sales are, by construction, those on which no method based on market evidence can succeed, and they are excluded from the comparable pool as an appraiser would exclude them; their inclusion as targets would raise the error of every method by a similar amount without affecting the comparisons, which are all paired on the same targets. A sensitivity analysis to the bounds of R8 and to the flip rule R9 is left for the next version of this paper.2526\subsection{Alternative accuracy measures}2728The MdAPE is our headline measure because the distribution of valuation errors is heavy-tailed and the median is the appraiser's natural loss. Table~\ref{tab:competition} shows that the MAPE and the shares within $\pm5$, $\pm10$ and $\pm20\%$ rank the models identically; the paired bootstrap on MAPE differences in Table~\ref{tab:pairs} yields the same conclusions as on MdAPE differences. The bias column shows that no comparable-based method is materially biased out of sample.2930\subsection{Index construction}3132The point-in-time index uses an expanding window of all prior sales and municipality fixed effects. Because comparables are at most 24 months old and the time adjustment contributes only \gainTime{}--\gainTimeLong{} points of accuracy (Section~\ref{sec:adjustments}), the comparable-based results are insensitive to the details of the index. The global benchmarks are more exposed: their level in 2025--2026 is carried entirely by the index from the last training quarter to the valuation month, and we verified that an index without municipality fixed effects, which is contaminated by the shift of sales towards less expensive municipalities, lowers their level.3334\subsection{Benchmark specification}3536The gradient-boosting benchmarks are trained with an L1 objective so that they estimate the conditional median, which is the quantity the MdAPE rewards; the same models trained with a squared-error objective have a median bias of about $-5\%$ in-sample and out of sample, because the distribution of log prices conditional on characteristics is left-skewed, and a correspondingly higher MdAPE; the direct hedonic model M10, estimated by least squares, is for the same reason recentred on the median of its validation residuals. The number of trees chosen by early stopping on the last 10\% of the training window is floored at 300 because the final, partial months of the sample make that slice unreliable. The hybrid weights of M12 and the reconciliation weights of M9 are convex combinations fitted on 2024 and applied unchanged; M9 places most of its weight on the assessment-ratio and GBM-adjusted indications, and M12 over 90\% of its weight on M11b.37