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\begin{appendices}

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\section{Additional Tables and Figures}
\label{app:tables}

% --- Appendix Table: Sub-sample Analysis ---
\begin{table}[htbp]
  \centering
  \begin{threeparttable}
    \caption{Sub-sample Analysis}
    \label{tab:app_subsample}
    \begin{tabular}{@{}lccc@{}}
      \toprule
      & (1) & (2) & (3) \\
      & Full Sample & Pre-2008 & Post-2008 \\
      \midrule
      $X_{i,t}$     & 0.172$^{***}$ & 0.195$^{***}$ & 0.148$^{***}$ \\
                     & (0.033)       & (0.048)       & (0.041)       \\[6pt]
      \midrule
      Controls       & Yes & Yes & Yes \\
      Firm FE        & Yes & Yes & Yes \\
      Time FE        & Yes & Yes & Yes \\[3pt]
      Observations   & 125{,}400 & 58{,}200 & 67{,}200 \\
      Adjusted $R^2$ & 0.190 & 0.175 & 0.203 \\
      \bottomrule
    \end{tabular}
    \begin{tablenotes}[flushleft]
      \footnotesize
      \item \textit{Notes:} This table reports sub-sample results. Column~(1) reproduces the baseline estimate from Table~\ref{tab:regression}, Column~(4). Columns~(2) and~(3) split the sample before and after January~2008. All specifications include the full set of controls and firm and time fixed effects. Standard errors clustered at the firm level are in parentheses. $^{***}$, $^{**}$, and $^{*}$ denote significance at the 1\%, 5\%, and 10\% levels.
    \end{tablenotes}
  \end{threeparttable}
\end{table}

% --- Appendix Table: Robustness Checks ---
\begin{table}[htbp]
  \centering
  \begin{threeparttable}
    \caption{Robustness Checks}
    \label{tab:app_robustness}
    \begin{tabular}{@{}lccc@{}}
      \toprule
      & (1) & (2) & (3) \\
      & Alt.\ Dep.\ Var. & Alt.\ Controls & Winsorized 5\% \\
      \midrule
      $X_{i,t}$     & 0.165$^{***}$ & 0.178$^{***}$ & 0.169$^{***}$ \\
                     & (0.036)       & (0.034)       & (0.032)       \\[6pt]
      \midrule
      Controls       & Yes & Yes & Yes \\
      Firm FE        & Yes & Yes & Yes \\
      Time FE        & Yes & Yes & Yes \\[3pt]
      Observations   & 125{,}400 & 125{,}400 & 119{,}800 \\
      Adjusted $R^2$ & 0.185 & 0.192 & 0.195 \\
      \bottomrule
    \end{tabular}
    \begin{tablenotes}[flushleft]
      \footnotesize
      \item \textit{Notes:} This table reports robustness checks. Column~(1) uses an alternative dependent variable. Column~(2) includes additional control variables. Column~(3) winsorizes all continuous variables at the 5th and 95th percentiles. All specifications include firm and time fixed effects. Standard errors clustered at the firm level are in parentheses. $^{***}$, $^{**}$, and $^{*}$ denote significance at the 1\%, 5\%, and 10\% levels.
    \end{tablenotes}
  \end{threeparttable}
\end{table}

\section{Variable Definitions}
\label{app:variables}

\begin{table}[htbp]
  \centering
  \caption{Variable Definitions}
  \label{tab:app_variables}
  \begin{tabular}{@{}p{3.5cm}p{10cm}@{}}
    \toprule
    Variable & Definition \\
    \midrule
    $r_{i,t}$        & Monthly log return of firm $i$ in month $t$, computed as $\ln(P_{i,t}/P_{i,t-1})$. \\[4pt]
    $\sigma_{i,t}$   & Monthly realized volatility of firm $i$, computed as the standard deviation of daily returns within month $t$, annualized by multiplying by $\sqrt{252}$. \\[4pt]
    $X_{i,t}$        & Main explanatory variable. [Provide specific definition.] \\[4pt]
    Size              & Natural logarithm of market capitalization (in millions of USD) at the end of month $t-1$. \\[4pt]
    Book-to-Market    & Ratio of book equity to market equity, using the most recent fiscal year-end book value and the market value at the end of June. Updated annually following \citet{fama1993common}. \\[4pt]
    Leverage          & Total debt divided by total assets, measured at the most recent fiscal year-end. \\[4pt]
    ROA               & Return on assets, defined as net income divided by total assets. \\
    \bottomrule
  \end{tabular}
\end{table}

\end{appendices}