\section{Simulation design and Monte Carlo results}\label{sec:simulation}

\subsection{Design}

Synthetic markets reproduce the features of the Quebec data that matter for index estimation (Appendix~\ref{app:sim}): 60 municipalities nested in 12 MRCs and 4 regions, expected monthly sales drawn from a log-normal distribution with a median of 3.3 sales (a quarter of the municipalities sell less than once a month, a few sell more than 50 times), latent quality-constant log price paths with nested random-walk deviations, a hedonic surface with a nonlinear floor-area effect and an optional drifting elasticity, spatially correlated cell quality, Student-$t$ noise with contamination, June--July seasonality in volumes, resales, missing floor areas and registration lags. Nine scenarios (Table~\ref{tab:mc_design}) vary the data-generating process: a baseline; a turning point at month 36; a flat market in which the floor area of sold homes trends upward (the median must rise spuriously); a rising market in which sold-home quality declines; drifting hedonic prices with increasing missingness; heavy tails (2.5 degrees of freedom, 4\% contamination); a sparse market with 35\% of the sales; and two deliberate counter-examples, a market without local dynamics (where pooling should dominate) and a market with strong idiosyncratic municipal dynamics (where pooling should hurt and a direct estimator should win). Each scenario is replicated 80 times with new seeds; every estimator of Section~\ref{sec:validation} that can be computed on a municipality is run on every cell, and SHARP-HPI is run in its full form and in five ablations (no empirical-Bayes pooling, no time-varying prices, local linear trend, Gaussian innovations, municipal-level location effects), in smoothed and filtered modes. The truth for a municipality is its latent path; for a region or the province it is the stock-weighted mean of its municipalities' paths. Reported metrics are averaged over cells and replications with Monte Carlo standard errors over replications (Tables~\ref{tab:mc_rmse_municipality}--\ref{tab:mc_lag}, Figures~\ref{fig:mc_rmse}--\ref{fig:mc_cov}).

\input{tables/tab_mc_design}

\subsection{Accuracy of monthly changes}

At the municipal level the ranking is unambiguous and stable across scenarios (Table~\ref{tab:mc_rmse_municipality}, Figure~\ref{fig:mc_rmse}). In the baseline the RMSE of the monthly log change is \MCmunicipalitySsharpbaselineRmse{} log points for SHARP-HPI, \MCmunicipalityKqhpibaselineRmse{} for the QHPI architecture (per-cell maximum-likelihood deviations), \MCmunicipalityCpooledtdbaselineRmse{} for a cell-level pooled time dummy, \MCmunicipalityBstratifiedbaselineRmse{} for the stratified median and \MCmunicipalityAmedianbaselineRmse{} for the median; repeat sales are estimable in only \MCmunicipalityHrepeatsalesbaselineShare\% of the municipal cell-months and are the noisiest where they exist. The composition scenarios show what quality adjustment buys: in the flat market with rising sold quality the median's monthly changes carry a bias of \MCmunicipalityAmedianflatqualityupBias{} log points per month (about 3\% per year of spurious appreciation), the hedonic estimators' \MCmunicipalitySsharpflatqualityupBias{}. The real-time (filtered) SHARP-HPI has an RMSE of \MCmunicipalitySsharpfilteredbaselineRmse{}, twice the smoothed value: half of the final accuracy at the municipal level comes from information that arrives after the month, which is the revision cost documented on the real data.

\input{tables/tab_mc_rmse_municipality}

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_mc_rmse.pdf}
\caption{Monte Carlo: RMSE of monthly log changes (log scale) at the municipality and region levels by scenario and method; whiskers are Monte Carlo standard errors.}\label{fig:mc_rmse}
\end{figure}

\subsection{What each component contributes}

The ablations identify the mechanisms. Empirical-Bayes calibration of the innovation variance improves the RMSE only slightly in the baseline (\MCmunicipalitySnoEBbaselineRmse{} without it) but matters for calibration: the 95\% intervals cover \MCmunicipalitySsharpbaselineCov\% of the true levels with it and \MCmunicipalitySnoEBbaselineCov\% without, and in the heavy-tailed scenario \MCmunicipalitySnoEBheavytailsCov\% against \MCmunicipalitySsharpheavytailsCov\%, because per-cell maximum likelihood collapses to the boundary in thin cells. The local-linear-trend variant is never better than the local level and doubles the revision between filtered and smoothed estimates (\MCmunicipalitySLLTbaselineRev{} against \MCmunicipalitySsharpbaselineRev{} log points). Gaussian innovations cost \MCmunicipalitySnorobustbaselineRmse{} against \MCmunicipalitySsharpbaselineRmse{} in the baseline and \MCmunicipalitySnorobustheavytailsRmse{} against \MCmunicipalitySsharpheavytailsRmse{} with heavy tails. Fine location effects matter in every scenario but one: replacing them by municipal effects raises the RMSE to \MCmunicipalityScoarselocbaselineRmse{} in the baseline. The exception is the heavy-tailed scenario, the first counter-example, where the municipal-location variant is the best estimator (\MCmunicipalityScoarselocheavytailsRmse{} against \MCmunicipalitySsharpheavytailsRmse): with 4\% contamination and 2.5 degrees of freedom, fine cells of fifteen sales absorb outliers into their fixed effects and transmit them to the residual means, and a coarser partition is more robust. Time-varying quality prices change nothing in the baseline and help marginally when prices drift.

\subsection{Counter-examples and the regional level}

The second counter-example, strong idiosyncratic municipal dynamics, did not produce the reversal it was designed for at the municipal level: the smoothed SHARP-HPI remains the most accurate (\MCmunicipalitySsharpidiosyncratictownsRmse{} against \MCmunicipalityKqhpiidiosyncratictownsRmse{} for per-cell estimation) because the empirical-Bayes prior is wide enough to let volatile cells keep large innovation variances. The cost appears in real time: the filtered estimate lags the truth by \MCmunicipalitySsharpfilteredidiosyncratictownsLag{} months by cross-correlation, against \MCmunicipalitySsharpfilteredbaselineLag{} in the baseline, and its coverage falls to \MCmunicipalitySsharpfilteredidiosyncratictownsCov\%. The regional level (Table~\ref{tab:mc_rmse_region}) is where SHARP-HPI's design costs accuracy: with 60 to 1{,}000 sales a month per region, the QHPI architecture, which estimates each liquid region with its own hedonic coefficients and smooths lightly, has an RMSE of \MCregionKqhpibaselineRmse{} against \MCregionSsharpbaselineRmse{} for SHARP-HPI, whose regional path is the pooled provincial path plus a smoothed deviation; the Monte Carlo confirms the repeat-sales evidence of Section~\ref{sec:validation} that region-specific estimation is preferable where the data allow it, and the panel's regional series should be read with that caveat. At the provincial level all hedonic estimators agree within 0.1 log point and the medians are three times noisier (Table~\ref{tab:mc_rmse_province}); the stock-weighted bottom-up aggregate of municipal SHARP indices is as accurate as the direct provincial estimate.

\input{tables/tab_mc_rmse_region}
\input{tables/tab_mc_rmse_province}
\input{tables/tab_mc_coverage}
\input{tables/tab_mc_bias}
\input{tables/tab_mc_lag}

\begin{figure}[htbp]\centering
\includegraphics[width=\textwidth]{figures/fig_mc_coverage.pdf}
\caption{Monte Carlo: empirical coverage of nominal 95\% intervals for municipal index levels.}\label{fig:mc_cov}
\end{figure}