Two-stage least squares as minimum distance

The two-stage least-squares (2SLS) instrumental-variables (IV) estimator for the parameters in linear models with a single endogenous variable is shown to be identical to an optimal minimum-distance (MD) estimator based on the individual instrument-specific IV estimators. The 2SLS estimator is a linear combination of the individual estimators, with the weights determined by their variances and covariances under conditional homoskedasticity. It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic. This provides an intuitive interpretation of the Sargan test. The equivalence results also apply to the efficient two-step generalized method of moments and robust optimal MD estimators and criterion functions, allowing for general forms of heteroskedasticity. It is further shown how these results extend to the linear overidentified IV model with multiple endogenous variables.