% ============================================================================
\section{Methodology}
\label{sec:methodology}
% ============================================================================

\subsection{Empirical Model}
\label{subsec:model}

To test our hypotheses, we estimate the following baseline specification:
%
\begin{equation}
  y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 Z_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t},
  \label{eq:baseline}
\end{equation}
%
where $y_{i,t}$ is the dependent variable for firm~$i$ in period~$t$, $X_{i,t}$ is the main variable of interest, $Z_{i,t}$ is a vector of control variables, $\gamma_i$ represents firm fixed effects, $\delta_t$ captures time fixed effects, and $\varepsilon_{i,t}$ is the error term.

Our coefficient of interest is $\beta_1$. Under the null hypothesis that \ldots{}, we expect $\beta_1 = 0$. A positive (negative) estimate of $\beta_1$ would indicate \ldots{}

\subsection{Identification Strategy}
\label{subsec:identification}

The main identification challenge arises from \ldots{} To address potential endogeneity concerns, we employ \ldots{}

We cluster standard errors at the firm level to account for within-firm serial correlation, following \citet{petersen2009estimating}.

\subsection{Extended Specifications}
\label{subsec:extended}

We extend the baseline model in several directions. First, we allow for nonlinear effects:
%
\begin{equation}
  y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 X_{i,t}^2 + \boldsymbol{\beta}_3' \mathbf{Z}_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t}.
  \label{eq:nonlinear}
\end{equation}

Second, we examine heterogeneous effects by interacting $X_{i,t}$ with \ldots{}:
%
\begin{equation}
  y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 D_{i,t} + \beta_3 (X_{i,t} \times D_{i,t}) + \boldsymbol{\beta}_4' \mathbf{Z}_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t},
  \label{eq:interaction}
\end{equation}
%
where $D_{i,t}$ is an indicator variable equal to one if \ldots{} The coefficient $\beta_3$ captures the differential effect of $X$ for \ldots{}