Two Stage Least Squares

Two-stage least squares (2SLS) is the standard implementation of Instrumental Variables estimation. Stage 1: regress the endogenous regressor $X$ on the instrument $Z$ (plus all exogenous controls) to get the fitted values $\hat X$. Stage 2: regress $Y$ on $\hat X$ (plus the same controls). The coefficient on $\hat X$ in stage 2 is the IV estimate $\hat\beta_{\text{IV}}$ — it uses only the $Z$-driven slice of $X$'s variation, which is exogenous by construction.

When to use

2SLS is the default IV recipe in modern applied work. In R, ivreg(y ~ x + controls | z + controls, data = df) from the AER package runs both stages and reports correct standard errors automatically (don't run the two stages manually with lm() — the second-stage SEs will be wrong). 2SLS is consistent but biased in finite samples; the bias shrinks as $n \to \infty$ and as the first-stage F grows. Always report the first-stage F to demonstrate the instruments aren't weak.