\section{Benchmarks, validation, ablation and robustness}\label{sec:validation}

\subsection{Benchmarks at the provincial and regional levels}\label{sec:bench}

Table~\ref{tab:bench_prov} reports every benchmark at the provincial level: the cumulative log change from January 2021 to September 2026 and the standard deviation of monthly log changes, with wall-clock times. Figure~\ref{fig:provmethods} plots the series. Four facts stand out. First, the quality-adjusted methods agree on the magnitude of the house appreciation: pooled time dummies, rolling time dummies with any window and splice, hedonic imputation, GEKS, HMTS, the QHPI architecture and SHARP-HPI all lie between \ProvDrtdwOneThreemovementhouseTotal{} and \ProvGhmtswOneThreehouseTotal{} log points, against \ProvAmedianhouseTotal{} for the median and \ProvHrepeatsalescshouseTotal{} for Case--Shiller repeat sales. The median understates because the sold houses became smaller and older over the sample; repeat sales overstate because resold houses are renovated. Second, the chained imputation indices drift relative to their direct counterparts for condominiums and plexes (Laspeyres \ProvEhilaspeyreschainedcondoTotal{} against Paasche \ProvEhipaaschechainedcondoTotal{} for condominiums), the classical chain-drift of monthly chaining with per-period hedonic functions estimated on a few thousand sales. Third, methods differ far more in month-to-month volatility than in trend: the standard deviation of monthly changes is about 1.9 log points for the time-dummy family and SHARP, 2.8 for imputation and GEKS, 3--4 for the median and repeat sales, and 1.3--1.9 for HMTS, whose state-space smoothing of each base-period series is visible. Fourth, the gradient-boosting residual index, which forecasts each month's prices from models trained on earlier months, rises by only \ProvJgbmootresidualhouseTotal{} log points: a predictive model that learns location and time jointly absorbs part of the appreciation into its fitted surface, which is why predictive accuracy is not index accuracy. SHARP-HPI with and without time-varying quality prices differ by less than one log point in total at the provincial level; the drift matters below the province.

\input{tables/tab_benchmarks_province}

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_province_methods.pdf}
\caption{Provincial indices by method (2021 average $=100$), January 2021 to September 2026.}\label{fig:provmethods}
\end{figure}

Table~\ref{tab:bench_region} moves to the 16 administrative regions, where sample sizes range from 60 to 1{,}200 sales a month and the methods that estimate a separate hedonic function per region become unstable. The regional comparison has no truth; we report the volatility of each series and its root mean squared distance to the regional Case--Shiller index over 12-month changes. Three results are relevant to the design. The median and stratified median are two to three times as volatile as the hedonic methods and the farthest from repeat sales. Among hedonic methods, the QHPI architecture, which fits each liquid region with its own coefficients and smooths lightly, is the least volatile (\RegSdKqhpidirectlocaltdhouse{} for houses) and the closest to repeat sales (\RegRmseRSKqhpidirectlocaltdhouse); SHARP-HPI, whose regional path is the pooled provincial path plus a smoothed deviation, is less volatile than the pooled time dummy (\RegSdSsharphouse{} against \RegSdCpooledtdhouse) but not closer to repeat sales than it (\RegRmseRSSsharphouse{} against \RegRmseRSCpooledtdhouse). The reason is identifiable: with 400 to 1{,}200 sales a month the regional Kalman gain is close to one, so the SHARP regional path is essentially the pooled model's monthly regional mean, and the pooled model constrains quality prices to be common across regions. The Monte Carlo experiments of Section~\ref{sec:simulation}, where the truth is known, show that this costs little when quality prices are common and that regional estimates are then close to the truth; on the real data the repeat-sales arbiter, biased though it is, suggests that region-specific quality prices carry information at the regional level. Figure~\ref{fig:regmethods} shows six regions.

\input{tables/tab_benchmarks_region}

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_region_methods.pdf}
\caption{Regional house indices by method for six regions of decreasing liquidity (2021 $=100$).}\label{fig:regmethods}
\end{figure}

\subsection{Agreement with repeat sales}

Table~\ref{tab:repeat_sales} compares SHARP-HPI with Case--Shiller repeat-sales indices estimated on the same cells where at least 300 pairs exist. At the provincial level the two house series have a correlation of \RSCorrLevelprovincehouse{} in levels; across regions the median correlation is \RSCorrLevelregionhouse{} and across the 20 municipalities with enough pairs \RSCorrLevelmunicipalityhouse. The repeat-sales total change exceeds the hedonic one by a median of \RSGapTotalregionhouse{} log points for houses at the regional level and \RSGapTotalregioncondo{} for condominiums. The sign and the class pattern are the classical ones: repeat sales of houses embed renovations and the selection of resold dwellings \citep{clapp1991alltransactions,gatzlaffhaurin1997,nowaksmith2020}, which bias them upward relative to a constant-quality index, while condominiums are renovated less. Repeat sales are an independent arbiter of the movement, not the truth, and the Monte Carlo evidence is the only place where the truth is known.

\input{tables/tab_repeat_sales}

\subsection{Composition shock}

Table~\ref{tab:composition} and Figure~\ref{fig:composition} report a synthetic quality-composition shock: in four regions, the smallest 40\% of houses (by floor area) are removed from the sample from January 2024 onward, which raises the median price mechanically. The median index records a spurious 2023-to-2024 jump of \CompShockAmedianZeroSix{} log points in the Montréal region and \CompShockAmedianZeroThree{} in the Capitale-Nationale; the pooled time dummy moves by \CompShockCpooledtdZeroSix{} and \CompShockCpooledtdZeroThree, and SHARP-HPI by \CompShockSsharpZeroSix{} and \CompShockSsharpZeroThree. The hedonic controls remove the shock almost entirely; the residual effect is the extrapolation of the floor-area curve to a truncated support.

\input{tables/tab_composition}
\begin{figure}[htbp]\centering
\includegraphics[width=0.6\textwidth]{figures/fig_composition_shock.pdf}
\caption{Composition shock (houses, four regions): shift of the 2023-to-2024 change after removing the smallest 40\% of houses from January 2024.}\label{fig:composition}
\end{figure}

\subsection{Placebo}

When sale months are randomly permuted within each municipality (keeping composition, location and the municipality's sales calendar but destroying the time structure), the estimated municipal deviations should be flat. Table~\ref{tab:placebo} shows that they are: the median standard deviation of the placebo deviation path is below 1.6 log points at the municipal level and below 0.3 at the MRC level, and the share of placebo 12-month changes outside their 95\% band is \PlaceboFPhousemunicipality\% for houses and \PlaceboFPcondomunicipality\% for condominiums at the municipal level, below the nominal 5\%. At the regional level for houses the rate is \PlaceboFPhouseregion\%, above nominal: with 50 or more sales a month the permuted means are not independent across months because the permutation preserves within-municipality volumes, and the residual seasonal composition survives. Figure~\ref{fig:placebo} contrasts the placebo and actual distributions of 12-month municipal changes.

\input{tables/tab_placebo}
\begin{figure}[htbp]\centering
\includegraphics[width=0.6\textwidth]{figures/fig_placebo.pdf}
\caption{Placebo (months permuted within municipality) versus actual absolute 12-month changes of municipal house indices.}\label{fig:placebo}
\end{figure}

\subsection{Residual spatial dependence}

Moran's $I$ of the stage-one residuals on ten nearest neighbours, computed in June of each year province-wide and within the three largest cities (Table~\ref{tab:moran}, Figure~\ref{fig:moran}), is small: within cities it ranges between $-0.02$ and $0.04$ and is significant at the 5\% level in a minority of the months tested. The 500-m to 1-km location cells absorb most of the spatial structure; what remains is a second-order concern for the index, which averages residuals over cells, but it means that model-based intervals understate the variance in cells where the sold dwellings cluster.

\input{tables/tab_moran}
\begin{figure}[htbp]\centering
\includegraphics[width=0.6\textwidth]{figures/fig_moran.pdf}
\caption{Moran's $I$ (10 nearest neighbours) of stage-one house residuals, June of each year.}\label{fig:moran}
\end{figure}

\subsection{Bootstrap versus model-based intervals}

A cluster bootstrap that resamples fine-location cells with replacement and re-estimates both stages (40 draws, condominiums) gives the comparison in Table~\ref{tab:bootstrap}. For the province the bootstrap standard error of the log level is about half the model-based one: the model-based $\mu$ standard error from the weighted regression does not exploit the within-cell dependence that the bootstrap preserves, and is conservative. For municipalities with 200 to 600 sales the bootstrap is \BootRatioLevelMedian{} times the model-based standard error (median over the cells tested): the stage-two variances treat the monthly residual means as independent observations with known variance, and the bootstrap adds the variance of the hedonic surface and of the location effects. For the thinnest cells the ratio falls below one, because resampling cells rather than sales leaves a cell with three sales almost unchanged. We report the model-based intervals in the panel and this calibration table alongside them; a user who wants conservative intervals for mid-sized municipalities should inflate them by the tabulated ratio.

\input{tables/tab_bootstrap}

\subsection{Ablations on the Quebec data}

Table~\ref{tab:ablation_real} reports, for each class and level, how far each variant's municipal monthly changes depart from the full model and the out-of-time calibration of its one-step-ahead innovations. Removing the empirical-Bayes calibration (per-cell maximum likelihood $\tau^2$) changes house municipal monthly changes by \AblnoEBhousemunicipality{} log points on average and lowers the mean local-data weight, because many thin cells end at the lower boundary of $\tau^2$ and are frozen to their parent. The Gaussian variant and the municipality-level location variant change the series by \Ablnorobusthousemunicipality{} and \Ablcoarselochousemunicipality{} log points, the largest departures: outlying months and within-municipality location shifts both matter. Removing the MRC level changes municipal series by \AblskipMRChousemunicipality{} log points, which measures how much the administrative hierarchy itself contributes. The out-of-time innovation diagnostics are above one for every variant (\AblZfullhouse{} for the full model at the municipal level): the model's one-step predictive variances are too small by 20--40\% for houses, consistent with the bootstrap, and no variant closes the gap; the local-linear-trend variant is slightly worse than the local level.

\input{tables/tab_ablation_real}

\subsection{Sample screens}

Table~\ref{tab:trimming} shows that the provincial time-dummy path is insensitive to the sample screens: keeping the assessment-ratio outliers, dropping type conflicts or new constructions, applying a monthly 1--99\% price trim, or keeping only roll-classified sales changes the 2021--2026 house path by at most one log point at any month, and the Huber and ordinary least-squares versions differ by less than two log points in total. The robustness comes from the robust likelihood and the location controls, not from the screens.

\input{tables/tab_trimming}

\subsection{Real-time vintages and revisions}\label{sec:revisions}

Releases were simulated monthly from July 2022 to October 2026 for the province, the regions and the 64 municipalities with at least 1{,}500 sales, with the ten-day publication delay measured in Section~\ref{sec:data}; at each release both stages are re-estimated on the data available then. Two publication modes are compared. Under \emph{full re-estimation} the published value of a month is the smoothed estimate given the data available at the release, so history is revised; under the \emph{frozen first release} the value of a month is its one-sided filtered estimate and never changes except through stage-one re-estimation, which is the state-space analogue of a movement splice. Table~\ref{tab:revisions} reports, for full re-estimation, the mean absolute revision of the monthly log change between the $k$-th release and the final retrospective series; Table~\ref{tab:revisions_frozen} the same for the frozen policy; Table~\ref{tab:policies} the provincial policies; Figure~\ref{fig:revisions} the profile and the fan of provincial vintages.

At the provincial level the first-release monthly change is within \PolFirsthouse{} log points of the final value for houses (\PolFirstcondo{} for condominiums, \PolFirstplex{} for plexes), about a tenth of the typical monthly change (\PolMeanAbshouse{}), and the third release is no closer: the provincial revision comes entirely from stage-one re-estimation with more data, not from the latest month, so a frozen policy loses nothing at the province. At the regional and municipal levels the two modes differ. For houses the first-release municipal monthly change is revised by \RevFirsthousemunicipality{} log points under full re-estimation and the gap to the final series under the frozen policy is \FrozenFirsthousemunicipality; for condominiums \RevFirstcondomunicipality{} against \FrozenFirstcondomunicipality. The revision declines over the first six releases as the smoother incorporates later months and then stabilises at the stage-one component; twelve-month changes are revised by \RevFirstTwelvehousemunicipality{} log points at first release for houses. The frozen policy therefore costs about half a log point per month at the municipal level relative to publishing revised values, the price of never revising, and the revision-limited policy (freeze at the third release) captures most of the gain of full re-estimation. These magnitudes are the timeliness--accuracy trade-off that a publisher of municipal indices faces; they are measured on this data and would differ with a slower registry.

\input{tables/tab_revisions}
\input{tables/tab_revisions_frozen}
\input{tables/tab_policies}

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_revisions.pdf}
\caption{Left: mean absolute revision of the monthly change by release number, houses (solid: full re-estimation; dashed: frozen first release). Right: every monthly vintage of the provincial house index, July 2022 to October 2026 (the darkest line is the latest).}\label{fig:revisions}
\end{figure}

\subsection{External comparison}

Table~\ref{tab:external} compares the provincial and metropolitan SHARP series with the Teranet--National Bank and CREA indices, rebased to the 2021 average and compared on annual means to avoid the thin January 2021. In the Québec City metropolitan area the agreement is close: the SHARP house aggregate rises by \ExtSHARPhousestockweightedCREASingleFamilyNSAQubecCMASharp{} log points from 2021 to 2026 against \CreaSFQcYr{} for the CREA single-family benchmark and \TeranetQcYr{} for Teranet (all types), and the condominium aggregate by \ExtSHARPcondostockweightedCREAApartmentNSAQubecCMASharp{} against \CreaAptQcYr{} for CREA apartments. In Montréal the SHARP house aggregate rises by \ExtSHARPhousestockweightedCREASingleFamilyNSAMontralCMASharp{} against \CreaSFMtlYr{} for CREA and \TeranetMtlYr{} for Teranet, a gap of about ten log points, while condominiums are within five points of CREA apartments. Table~\ref{tab:external_rs} shows that the gap is in the data rather than in the estimator: repeat sales on our own Montréal house sample rise by \ExtRSMontralCMMhouses{} log points, the same as the SHARP aggregate, whereas repeat sales on all three classes together rise by \ExtRSMontralCMMallthreeclasses{}, closer to Teranet's all-type \TeranetMtlYr. Two explanations remain for the house gap relative to CREA: the CREA benchmark uses bedroom, bathroom and other listing attributes that absorb quality upgrades our roll variables cannot, and brokered resales are a different population from registry deeds. The correlation of 12-month changes between SHARP and the external series is between \ExtSHARPcondostockweightedTeranetNSArepeatsalesMontralCMACorr{} and \ExtSHARPhousestockweightedTeranetNSArepeatsalesMontralCMACorr{} in Montréal; in Québec City it is lower because the SHARP metropolitan aggregate is noisier month to month than the smoothed external series.

\input{tables/tab_external}
\input{tables/tab_external_rs}