First Stage

In 2SLS, the first stage is the regression of the endogenous regressor $X$ on the instrument(s) $Z$ and all exogenous controls: $X_i = \gamma_0 + \gamma_1 Z_i + \boldsymbol\gamma_2' W_i + v_i$. Its fitted values $\hat X_i$ are what enter the second-stage outcome equation. The first-stage F-statistic on the instruments is the standard diagnostic for relevance / weak instruments.

When to use

Always report the first stage alongside the second — readers need to see that the instruments actually move $X$. A first-stage coefficient of the expected sign and magnitude is reassuring; a tiny coefficient or low F is a red flag. R: with ivreg, call summary(model, diagnostics = TRUE) to see the first-stage F.