Second Stage

In 2SLS, the second stage is the regression of the outcome $Y$ on the fitted endogenous regressor $\hat X$ (from the First Stage) and the exogenous controls: $Y_i = \beta_0 + \beta_1 \hat X_i + \boldsymbol\beta_2' W_i + u_i$. The coefficient $\hat\beta_1$ is the IV estimate of the causal effect of $X$ on $Y$.

When to use

Mechanically, the second stage is just another OLS regression — but never run it by hand with lm(Y ~ X_hat), because the SEs ignore the uncertainty in $\hat X$ from the first stage and will be wrong. Use ivreg() or fixest::feols() with IV syntax, which compute the correct asymptotic SEs.