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\section{Robustness Checks}
\label{sec:robustness}
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We conduct several robustness tests to assess the sensitivity of our baseline results.

\subsection{Alternative Variable Definitions}
\label{subsec:alt_definitions}

We first verify that our results are not driven by the specific construction of the dependent variable. Table~\ref{tab:app_robustness} in Appendix~\ref{app:tables} reports estimates using \ldots{} as alternative measures. The coefficient on $X_{i,t}$ remains statistically significant and economically meaningful across all specifications.

\subsection{Alternative Estimation Methods}
\label{subsec:alt_estimation}

We re-estimate the model using \ldots{} to address potential concerns about \ldots{} The results are qualitatively unchanged.

\subsection{Placebo Tests}
\label{subsec:placebo}

As a falsification exercise, we replace the main explanatory variable with \ldots{} If our baseline results reflect a genuine economic relationship rather than spurious correlation, we expect the placebo coefficients to be statistically insignificant. Consistent with this expectation, none of the placebo estimates are significant at conventional levels.

\subsection{Sensitivity to Sample Period}
\label{subsec:sample_period}

We examine whether our results are sensitive to the sample period by \ldots{} The estimates remain stable across sub-periods, suggesting that our findings are not driven by a specific time window.