Two-Stage Least Squares with Robust Inference

Replacing endogenous variation

An instrument $z$ must predict the endogenous regressor $x$ and remain excluded from the structural error. Two-stage least squares projects $x$ onto the instrument space (Angrist and Pischke, 2009).

A Wald-ratio example

With one binary instrument and one endogenous regressor, the slope can be written as a ratio of group differences.

The reduced-form difference is $6$. The first-stage difference is $3$. The IV estimate is $6/3=2$.

Assumptions

Relevance requires a nonzero first stage. Exogeneity requires the instrument to be orthogonal to the structural error. Exclusion requires no outcome path outside the endogenous variable. In causal settings, monotonicity supports a local average treatment effect interpretation.

Weak instruments can make conventional normal approximations unreliable. Overidentification does not prove instrument validity.

Robust covariance

Heteroskedasticity-robust IV covariance uses structural residuals and projected regressors. HC0 and HC1 differ by a finite-sample scale. Software also differs in whether it applies this scale by default.

The first-stage projection splits $x$ into instrument-predicted and residual components. Only the predicted component identifies the second-stage coefficient.

What the IV coefficient represents

Two-stage least squares uses the part of $x$ predicted by the excluded instruments. In the just-identified binary case, the coefficient equals the reduced-form outcome effect divided by the first-stage treatment effect. Its causal meaning requires instrument relevance, exclusion, independence, and a suitable treatment-response framework. With heterogeneous effects and monotonic treatment selection, the estimate can describe a local average treatment effect for compliers.

Report the first-stage coefficient, reduced form, 2SLS estimate, and robust confidence interval. State which regressors are endogenous and which instruments are excluded. A large 2SLS coefficient can arise from a small first stage, so the denominator needs equal attention.

Identification and robust inference

Construct the instrument matrix with every included exogenous regressor plus the excluded instruments. Verify matrix rank and use structural residuals in the heteroskedasticity-robust covariance. The first-stage fitted values define the coefficient calculation. Regressing the outcome directly on fitted values with ordinary second-stage standard errors gives incorrect uncertainty.

Inspect first-stage strength and report a weak-instrument diagnostic suited to the model. When strength is limited, use Anderson-Rubin or other weak-IV-robust confidence sets (Andrews et al., 2019). Robust covariance handles heteroskedasticity. It does not address weak identification, invalid exclusion, or clustered errors. Cluster the score contributions when instrument and outcome shocks share group-level dependence. Discuss the substantive path from each instrument to the outcome because overidentification tests cannot verify exclusion by themselves.

Further reading

Angrist and Pischke give an applied treatment of IV and local effects (Angrist and Pischke, 2009). Andrews, Stock, and Sun review weak-instrument inference (Andrews et al., 2019).