% Author: Simon-Pierre Boucher — contact@spboucher.ai
% ============================================================================
\section{Measurement-error bounds and first stage}
\label{app:bounds}

This appendix derives the bounds used in Section~\ref{sec:meas} and
reports the moments behind them. Write the within-cell demeaned variables
as in equation~\eqref{eq:me}, $\ln SP_i = \ln V_i + u_i$ and $\ln AV_i =
\beta \ln V_i + \varepsilon_i$, with $u_i$, $\varepsilon_i$ and $\ln V_i$
mutually uncorrelated and $\sigma^2_V$, $\sigma^2_u$, $\sigma^2_\varepsilon$
their variances. The observable within-cell moments are
\begin{equation}
\operatorname{Var}(\ln SP) = \sigma^2_V + \sigma^2_u, \qquad
\operatorname{Var}(\ln AV) = \beta^2\sigma^2_V + \sigma^2_\varepsilon, \qquad
\operatorname{Cov}(\ln AV, \ln SP) = \beta\,\sigma^2_V .
\label{eq:moments}
\end{equation}
The forward slope is $\hat\beta_F = \operatorname{Cov}/\operatorname{Var}(\ln
SP) \to \beta\sigma^2_V/(\sigma^2_V+\sigma^2_u) \le \beta$, and the inverse
of the reverse slope is $1/\hat\beta_R = \operatorname{Var}(\ln
AV)/\operatorname{Cov} \to \beta + \sigma^2_\varepsilon/(\beta\sigma^2_V)
\ge \beta$, which gives equation~\eqref{eq:bounds}. Given $\hat\beta_F$,
the noise share $\lambda = \sigma^2_u/\operatorname{Var}(\ln SP)$ that would
rationalise a true elasticity $\beta^\ast$ is $\lambda = 1 -
\hat\beta_F/\beta^\ast$, and the implied standard deviation of transaction
noise is $\sqrt{\lambda\operatorname{Var}(\ln SP)}$. Table~\ref{tab:bounds}
reports the moments, the bounds, the noise shares implied by $\beta^\ast =
1$ and by the Clapp estimate, and the first stage of the Clapp instrument.
Three features deserve comment. The bracket $[0.656, 1.137]$ contains one,
so the classical errors-in-variables model alone cannot establish
regressivity on the baseline sample; this is a property of the large
assessment-error variance (the within-cell correlation is 0.76), not of
the sample size. Rationalising proportionality requires a transaction-noise
standard deviation of 0.28 log points, and the Clapp estimate itself
implies 0.25; both exceed the out-of-sample dispersion of hedonic
predictions around prices on comparable Quebec data, which suggests that
part of the ``noise'' the instrument discounts is non-market sales rather
than bargaining dispersion. The first stage is strong by any standard, so
the instrument's limitation is the dependence of $Z$ on the assessment
rank discussed in Section~\ref{sec:meas}, not weak identification.

\begin{table}[h]
\centering
\begin{threeparttable}
\caption{Within-cell moments, measurement-error bounds and Clapp first stage, baseline sample}
\label{tab:bounds}
\small
\input{../results/tables/rev_bounds}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes:} All moments are computed on variables demeaned
within the 2{,}884 municipality~$\times$~roll~$\times$~sale-year cells.
The first-stage regression is of the demeaned $\ln SP$ on $Z \in
\{-1,0,+1\}$ with municipality-clustered standard errors; $F$ is the
square of the cluster-robust $t$-statistic.
\end{tablenotes}
\end{threeparttable}
\end{table}