spb/wp22_uqo

1\section{Estimand and measurement challenges}\label{sec:estimand}23\subsection{Definition}45Let $g$ index a geography in the hierarchy (province, region, MRC, municipality), $k \in \{\text{house}, \text{condominium}, \text{plex}\}$ a property class and $t$ a month. Let $\mathcal{X}$ be the space of observable quality and location attributes and $F_{g,k}$ a fixed reference distribution of attributes for the pair $(g,k)$. The object of measurement is the quality-constant log price level6\begin{equation}7L_{g,k,t} \;=\; \int_{\mathcal{X}} \E\!\left[\log P \mid x, g, k, t\right] \, dF_{g,k}(x),8\label{eq:estimand}9\end{equation}10and the index is $I_{g,k,t} = 100 \exp\!\left(L_{g,k,t} - \bar L_{g,k,2021}\right)$, where $\bar L_{g,k,2021}$ is the average over the months of 2021 (the primary normalisation); January 2021 $=100$ is offered as a secondary normalisation. The base does not enter any growth rate: monthly and annual log changes $\Delta L_{g,k,t}$ and $\Delta_{12} L_{g,k,t}$ are invariant to it, which is verified in the code tests. Equation~\eqref{eq:estimand} is a price index for a \emph{constant basket}, not for the stock or for the transacted properties; the stock enters only through aggregation weights (Section~\ref{sec:validation}), and representative dollar prices are published separately as the exponential of the mean log price of the cell's sample at base-period prices, carried forward by the index.1112\subsection{Identification in a hedonic model}1314With a log-linear hedonic specification $\E[\log P \mid x, g, k, t] = f_k(x; \theta_t) + \alpha_{\ell(x)} + L_{g,k,t}$, the level $L_{g,k,t}$ is identified only up to the location of the reference distribution: a change in which dwellings sell changes the sample mean of $f_k(x;\theta_t) + \alpha_\ell$ and must not be read as a price change. Three design choices follow. First, location enters through fixed effects at a scale fine enough that the composition of sales within a cell is approximately constant in the attributes that are not observed; the coarser the cell, the more unobserved location quality leaks into the index. Second, the quality prices $\theta_t$ are allowed to drift smoothly: a floor-area elasticity that changes from 0.42 to 0.30 over five years (Table~\ref{tab:stage1}) would otherwise bias the index for cells whose sold dwellings are larger or smaller than the provincial reference. Third, the reference distribution is the cell's own sample over the whole period, so that the index is a within-cell comparison; common-support rules record the share of a cell's sales inside the central 98\% of the class's floor-area and age distributions.1516\subsection{Measurement challenges specific to thin markets}1718A municipality with four sales a month yields a monthly mean residual with a standard error of roughly $0.19/\sqrt{4} \approx 0.10$ log points for houses, an order of magnitude larger than any plausible monthly price change. Any estimator that reports a monthly change from such a cell without pooling is reporting noise, and any estimator that pools without saying how much is hiding the problem. The framework therefore reports, for every cell-month, the actual number of sales, the effective sample size implied by the posterior variance, the share of the estimate that comes from the local data (the Kalman gain), a 95\% interval, and a grade. A model estimate for a month with zero local sales is a model-assisted interpolation and is labelled as such; cells whose effective sample size falls below five are labelled insufficiently identified and suppressed in the headline tables while remaining in the panel for transparency.1920Two further challenges are specific to the data. The publication delay is short and backfill is absent, so real-time incompleteness is confined to the last month; but the roll's single vintage means that the characteristics are post-sale, and the \$50{,}000 floor truncates the price distribution in the cheapest markets. Both are discussed where they matter.21