In GMM estimation it is well known that if the number of moment conditions grows with the sample size, GMM asymptotics differ from the standard case with moment size fixed as the sample size tends to infinity. The present work explores infinite dimensional GMM estimation under various conditions on the moment conditions and the weight matrix. Our approach employs a partial sum process formed by the moment conditions to represent high dimensional moments and an invariance principle to capture the infinite dimensional asymptotics as the moment size grows. Next, the GMM weight matrix is assumed to converge to one of two kernels at the limit: a continuous kernel or the Dirac delta function. Combining these different conditions enables development of a large sample theory for most efficient GMM estimation. The effects of permuting the moment conditions on GMM efficiency are also explored. The resulting theory is applied to weak instrumental variable estimation and the Angrist and Krueger (1991) data are re-analyzed in an empirical application of the new methods.
