spb/wp10_uqo

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2% ============================================================================3\section{Measurement-error bounds and first stage}4\label{app:bounds}56This appendix derives the bounds used in Section~\ref{sec:meas} and7reports the moments behind them. Write the within-cell demeaned variables8as in equation~\eqref{eq:me}, $\ln SP_i = \ln V_i + u_i$ and $\ln AV_i =9\beta \ln V_i + \varepsilon_i$, with $u_i$, $\varepsilon_i$ and $\ln V_i$10mutually uncorrelated and $\sigma^2_V$, $\sigma^2_u$, $\sigma^2_\varepsilon$11their variances. The observable within-cell moments are12\begin{equation}13\operatorname{Var}(\ln SP) = \sigma^2_V + \sigma^2_u, \qquad14\operatorname{Var}(\ln AV) = \beta^2\sigma^2_V + \sigma^2_\varepsilon, \qquad15\operatorname{Cov}(\ln AV, \ln SP) = \beta\,\sigma^2_V .16\label{eq:moments}17\end{equation}18The forward slope is $\hat\beta_F = \operatorname{Cov}/\operatorname{Var}(\ln19SP) \to \beta\sigma^2_V/(\sigma^2_V+\sigma^2_u) \le \beta$, and the inverse20of the reverse slope is $1/\hat\beta_R = \operatorname{Var}(\ln21AV)/\operatorname{Cov} \to \beta + \sigma^2_\varepsilon/(\beta\sigma^2_V)22\ge \beta$, which gives equation~\eqref{eq:bounds}. Given $\hat\beta_F$,23the noise share $\lambda = \sigma^2_u/\operatorname{Var}(\ln SP)$ that would24rationalise a true elasticity $\beta^\ast$ is $\lambda = 1 -25\hat\beta_F/\beta^\ast$, and the implied standard deviation of transaction26noise is $\sqrt{\lambda\operatorname{Var}(\ln SP)}$. Table~\ref{tab:bounds}27reports the moments, the bounds, the noise shares implied by $\beta^\ast =281$ and by the Clapp estimate, and the first stage of the Clapp instrument.29Three features deserve comment. The bracket $[0.656, 1.137]$ contains one,30so the classical errors-in-variables model alone cannot establish31regressivity on the baseline sample; this is a property of the large32assessment-error variance (the within-cell correlation is 0.76), not of33the sample size. Rationalising proportionality requires a transaction-noise34standard deviation of 0.28 log points, and the Clapp estimate itself35implies 0.25; both exceed the out-of-sample dispersion of hedonic36predictions around prices on comparable Quebec data, which suggests that37part of the ``noise'' the instrument discounts is non-market sales rather38than bargaining dispersion. The first stage is strong by any standard, so39the instrument's limitation is the dependence of $Z$ on the assessment40rank discussed in Section~\ref{sec:meas}, not weak identification.4142\begin{table}[h]43\centering44\begin{threeparttable}45\caption{Within-cell moments, measurement-error bounds and Clapp first stage, baseline sample}46\label{tab:bounds}47\small48\input{../results/tables/rev_bounds}49\begin{tablenotes}[flushleft]\footnotesize50\item \textit{Notes:} All moments are computed on variables demeaned51within the 2{,}884 municipality~$\times$~roll~$\times$~sale-year cells.52The first-stage regression is of the demeaned $\ln SP$ on $Z \in53\{-1,0,+1\}$ with municipality-clustered standard errors; $F$ is the54square of the cluster-robust $t$-statistic.55\end{tablenotes}56\end{threeparttable}57\end{table}58