IV2SLS

IV2SLS, or instrumental variables two-stage least squares, is an estimation method used in econometrics to estimate linear regression models when one or more regressors are endogenous. The typical model is y = Xβ + ε, where y is the dependent variable, X contains endogenous regressor(s), β are coefficients, and ε is the error term. Valid instruments Z are correlated with the endogenous regressors but uncorrelated with the error term, and exogenous controls W may be included.

The estimation proceeds in two stages. In the first stage, each endogenous regressor is regressed on the instruments Z and the exogenous controls W to obtain fitted values X̂. In the second stage, y is regressed on X̂ (and W) by ordinary least squares to obtain β̂. The IV2SLS estimator is consistent provided the instruments are exogenous (E[Z′ε] = 0) and sufficiently relevant, meaning they are correlated with the endogenous regressors.

Identification depends on the number of instruments L relative to the number of endogenous regressors K. If L = K, the model is exactly identified; if L > K, it is overidentified, allowing tests of instrument validity such as Hansen’s J test. Weak instruments—where instruments poorly predict the endogenous regressors—can lead to biased and imprecise estimates, making the first-stage F-statistic a common diagnostic.

Standard errors in IV2SLS can be computed under homoskedasticity, but robust or heteroskedasticity-consistent methods are often used. Related approaches include LIML and efficient GMM, as well as system estimators like 3SLS for multiple equations. The causal interpretation of IV2SLS relies on the validity of the exclusion restrictions and the strength of the instruments.