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2.7 To GMM or not to GMM?
The advantages of GMM over IV are clear: if heteroskedasticity is present, the GMM
estimator is more ecient than the simple IV estimator, whereas if heteroskedasticity
is not present, the GMM estimator is no worse asymptotically than the IV estimator.
Nevertheless, the use of GMM does come with a price. The problem, as Hayashi
(2000) points out (p. 215), is that the optimal weighting matrix ^ S at the core of ecient
GMM is a function of fourth moments, and obtaining reasonable estimates of fourth
moments may require very large sample sizes. The consequence is that the ecient
GMM estimator can have poor small sample properties. In particular, Wald tests tend
to over{reject the null (good news for the unscrupulous investigator in search of large
t statistics, perhaps, but not for the rest of us). If in fact the error is homoskedastic,
IV would be preferable to ecient GMM. For this reason a test for the presence of
heteroskedasticity when one or more regressors is endogenous may be useful in deciding
whether IV or GMM is called for. Such a test was proposed by Pagan and Hall (1983)
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