Asymptotic Equivalence of Closest Moments and GMM Estimators

This note considers an asymptotic property of the class of closest moments estimators.
Each such estimator is obtained by setting a vector of sample moments
close to corresponding population moments. It is shown that each such estimator
is asymptotically equivalent to a GMM estimator, which has a quadratic
distance function. An implication of this result is that the estimator that is
asymptotically efficient in the GMM class is also asymptotically efficient in the
wider class of closest moment estimators.

Method-of-moments estimators have long been of interest, as has been the
issue of efficient estimation when there are fewer parameters than moments.
For the classical method of moments, Chiang [5] showed that the classical
minimum chi-square estimator is efficient in the class of estimators that are
smooth functions of the sample moments and nuisance parameter estimates.
Recently, Hansen [6] has formulated a GMM (generalized method of moments)
class of estimators that includes cases where the sample moments depend
on parameters. A GMM estimator is the parameter value that minimizes
a quadratic form in the difference of population and sample moments. This
class of estimators, which includes instrumental variable estimators, has long
been of interest in econometrics. Hansen [6] showed that the generalization
of the classical minimum chi-square estimator, which is obtained by choosing
the matrix in the quadratic form to be a consistent estimator of the inverse
asymptotic covariance matrix of the moments, is asymptotically efficient in
the class of GMM estimators. More recently, Chamberlain [4] has shown
that in an i.i.d. (independent and identically distributed) environment the
optimal GMM estimator is asymptotically efficient among all estimators that
use only the information provided by the moments.