GMM inference when the number of moment conditions is large

Asymptotic theory typically presumes that the dimensionality of econometric models is independent of the sample size even though this presumption is often quite unrealistic. In GMM estimation, whenever optimal instruments are not available, it can frequently be shown that adding over-identifying restrictions (moment conditions) will increase asymptotic precision. However, the conventional asymptotics which underlies this view insists that the number of moment conditions remain finite even though the number of available moment conditions may grow without bound. We consider the explicit dependence of the number of moment conditions (or instruments), qn, on the sample size, n, and establish that, under conventional regularity conditions for the estimation of a linear model with general heteroskedasticity, qn3/n→0 is a sufficient condition for the validity of conventional asymptotic inference about the GMM estimator.