spb/wp22_uqo

1\section{Quebec monthly indices, 2021--2026}\label{sec:empirical}23\subsection{Stage one: quality prices}45Table~\ref{tab:stage1} reports the stage-one regressions. The within-cell fit is \RtwoWithinhouse{} for houses, \RtwoWithincondo{} for condominiums and \RtwoWithinplex{} for plexes, with robust residual standard deviations of \SigmaRobusthouse, \SigmaRobustcondo{} and \SigmaRobustplex{} log points. Floor area carries the expected convexity (Figure~\ref{fig:hedonic}), age profiles show a premium for near-new dwellings and depreciation that flattens after sixty years for houses and reverses for old condominiums in central locations, and the time interactions are negative for floor area in all three classes: the elasticity of price with respect to floor area fell during the sample (\BetaFaTThouse{} per unit of the centred time index for houses), which is consistent with smaller dwellings appreciating faster than larger ones. A fixed-coefficient model (the ``no time-varying prices'' ablation) changes municipal monthly changes by \AblnoTVhousemunicipality{} log points on average, little, but the drift is real and costless to allow.67\input{tables/tab_stage1}89\begin{figure}[htbp]\centering10\includegraphics[width=0.95\textwidth]{figures/fig_hedonic_profiles.pdf}11\caption{Stage-one quality profiles: age effects (left) and floor-area effects (right), by class.}\label{fig:hedonic}12\end{figure}1314\subsection{Stage two: calibrated signal-to-noise ratios}1516Table~\ref{tab:priors} reports the empirical-Bayes priors. The prior median innovation standard deviation of regional deviations is \PriorTauhouseregion\% per month for houses with a tight prior variance (\PriorVhouseregion), that of municipalities \PriorTauhousemunicipality\% with a wide prior (\PriorVhousemunicipality): municipalities differ genuinely in how much they move relative to their MRC, and the calibration lets liquid cells keep their own estimate while thin cells inherit the typical ratio. Figure~\ref{fig:gain} shows the consequence: the mean local-data weight rises with liquidity from near zero below one sale a month to above 0.8 in the largest cities, and the effective sample size tracks the actual count for liquid cells and exceeds it for thin cells, which is the pooling.1718\input{tables/tab_priors}1920\begin{figure}[htbp]\centering21\includegraphics[width=0.95\textwidth]{figures/fig_gain_vs_n.pdf}22\caption{Municipal cells: mean local-data weight (Kalman gain) and mean effective sample size against median monthly sales.}\label{fig:gain}23\end{figure}2425\subsection{The panel}2627The panel contains \NCells{} geography--class cells (\NCellsMun{} municipal) over 69 months. The provincial house index stands at \ProvSephouse{} in September 2026 (2021 average $=100$), condominiums at \ProvSepcondo{} and plexes at \ProvSepplex{}, with twelve-month changes of \ProvYoySephouse\%, \ProvYoySepcondo\% and \ProvYoySepplex\%. Table~\ref{tab:top_mun} gives the twelve largest municipalities with intervals and grades, Figure~\ref{fig:munex} six municipalities spanning the liquidity range, and Figure~\ref{fig:map} the map of the municipal house index with the reliability hatching. Across the \NMunHouseABCSep{} municipalities graded A to C for houses in September 2026, the index ranges from \MunHousePTen{} at the tenth percentile to \MunHousePNinety{} at the ninetieth (median \MunHousePFifty): the dispersion of five-year appreciation across Quebec municipalities is as large as the provincial appreciation itself. Table~\ref{tab:grades} gives the distribution of grades: most regional and MRC cell-months are graded A to C, while a majority of municipal cell-months are graded D or E and are published for transparency only. The series are not seasonally adjusted; the monthly profile visible in Figure~\ref{fig:munex} (dips in winter months) is a composition effect in unobserved attributes that the hedonic controls do not remove and that users of monthly changes should treat as seasonal.2829\input{tables/tab_top_municipalities}30\input{tables/tab_grades}3132\begin{figure}[htbp]\centering33\includegraphics[width=\textwidth]{figures/fig_municipal_examples.pdf}34\caption{SHARP-HPI house indices with 95\% intervals for six municipalities of decreasing liquidity (smoothed, retrospective mode).}\label{fig:munex}35\end{figure}3637\begin{figure}[htbp]\centering38\includegraphics[width=\textwidth]{figures/fig_map_municipal_house.pdf}39\caption{Municipal house index in September 2026 (2021 $=100$) on 2026 municipal boundaries; hatched cells are graded D or E.}\label{fig:map}40\end{figure}4142\subsection{Aggregation and coherence}4344Section~\ref{sec:estimand} defined the headline as a constant-basket index. The provincial path $\mu_{k,t}$ is estimated on all sales and is therefore transaction-weighted across municipalities. Table~\ref{tab:aggregation} compares it with bottom-up aggregates of the municipal indices weighted by the housing stock (dwelling units or assessed value, from the roll), by transactions, and equally. The stock-weighted house aggregate differs from the direct provincial path by \AgghousestockunitsGap{} log points over 2021--2026 (root mean squared gap \AggRmsehousestockunits{} per month) and the equal-weighted aggregate by \AgghouseeqGap{}: small municipalities appreciated more than large ones, so the choice of weights is a choice of estimand, not a reconciliation error. A reconciliation projection that forces the direct and bottom-up series to agree would move the provincial series toward whichever weighting is imposed; we do not impose it and report the gap instead.4546\input{tables/tab_aggregation}47