Abstract
SummaryThe 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.
Highlights
For a single endogenous variable linear model with multiple instruments, the standard instrumental variables (IV) estimator is the Two-Stage Least Squares (2SLS) estimator, which is a consistent and asymptotically e¢ cient estimator under standard regularity assumptions and conditional homoskedasticity, see e.g. Hayashi (2000, p 228)
It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the Minimum Distance (MD) criterion test statistic, providing another intuitive interpretation of the Sargan test
It appears that these equivalence results are not available in the literature, and are not discussed in standard textbooks
Summary
For a single endogenous variable linear model with multiple instruments, the standard IV estimator is the Two-Stage Least Squares (2SLS) estimator, which is a consistent and asymptotically e¢ cient estimator under standard regularity assumptions and conditional homoskedasticity, see e.g. Hayashi (2000, p 228). It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic, providing another intuitive interpretation of the Sargan test It appears that these equivalence results are not available in the literature, and are not discussed in standard textbooks. If the instruments are denoted by z1, z2, z3, and z4, the collection of sets f(z1; z2) ; (z2; z3) ; (z3;z4)g is su¢ cient This results in three just identi...ed IV estimates of the two parameters of interest, and Section 3 shows that the per parameter optimal minimum distance estimators are identical to the 2SLS estimators
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