\begin{table}[!ht]
\caption{Implied effect of the focal's fiscal expansion on the neighbour (Wald/IV ratios)}\label{table:ivtable}
\centering
\begin{tabular}{lccc}\hline\hline
 & Reduced form & First stage & Implied IV \\
Neighbour outcome (log) & crossing $\to$ Y & crossing $\to$ focal exp. & Y per 1\% focal exp. \\ \hline
Total expenditure         &  1.09\phantom{**} (0.69)   & 6.24 (0.63) &  0.18\phantom{**} (0.11)   \\
Public-sector employment  &  2.67**         (1.29)     & 6.24 (0.63) &  0.43**         (0.21)     \\
Number of firms           & $-1.56$\phantom{**} (1.00) & 6.24 (0.63) & $-0.25$\phantom{**} (0.16) \\
Private-sector employment & $-1.05$\phantom{**} (1.53) & 6.24 (0.63) & $-0.17$\phantom{**} (0.25) \\
\hline\hline
\end{tabular}\par
\begin{tablenotes}\footnotesize
\item All entries are overall Callaway--Sant'Anna ATTs in percent (log coefficients $\times 100$), with neighbour-clustered influence-function standard errors in parentheses. The \emph{reduced form} is the effect of the focal's FPM threshold crossing on the neighbour outcome; the \emph{first stage} is its effect on the focal's total expenditure (common across rows). The \emph{implied IV} is the Wald ratio of the two --- the neighbour response per one-percent increase in focal expenditure, under the exclusion restriction that the crossing affects the neighbour only through the focal's fiscal expansion --- with a delta-method standard error treating the two estimates as independent. Because the reduced form and the first stage are computed on the same sample and event timing within a common influence-function framework, their sampling covariance is not exactly zero, so this standard error is approximate; the first stage is, however, estimated precisely ($t\approx 10$) and is common across rows, so the variance of the ratio is dominated by the reduced-form term and the omitted covariance contributes little in practice. The exclusion restriction is strong (it rules out, for instance, a direct cross-border labour-market channel), so the implied-IV column is best read as a descriptive rescaling of the reduced form rather than as a structural parameter. Consistent with the robustness evidence in Section~\ref{sec:identification}, only the public-sector response is statistically distinguishable from zero; the negative private-side ratios are imprecise. $^{*}\,p<0.10$, $^{**}\,p<0.05$, $^{***}\,p<0.01$.
\end{tablenotes}\end{table}