Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators

In an effort to improve the small sample properties of generalized method of moments
(GMM) estimators, a number of alternative estimators have been suggested.
These include empirical likelihood (EL), continuous updating, and exponential tilting
estimators. We show that these estimators share a common structure, being members
of a class of generalized empirical likelihood (GEL) estimators. We use this structure
to compare their higher order asymptotic properties. We find that GEL has no asymptotic
bias due to correlation of the moment functions with their Jacobian, eliminating
an important source of bias for GMM in models with endogeneity. We also find that EL
has no asymptotic bias from estimating the optimal weight matrix, eliminating a further
important source of bias for GMM in panel data models. We give bias corrected GMM
and GEL estimators. We also show that bias corrected EL inherits the higher order
property of maximum likelihood, that it is higher order asymptotically efficient relative
to the other bias corrected estimators.
KEYWORDS:GMM, empirical likelihood, bias, higher order efficiency, stochastic expansions.

1. INTRODUCTION
IN AN EFFORT TO IMPROVE the small sample properties of GMM, a number
of alternative estimators have been suggested. These include the empirical
likelihood (EL) estimator of Owen (1988), Qin and Lawless (1994), and
Imbens (1997), the continuous updating estimator (CUE) of Hansen, Heaton,
and Yaron (1996), and the exponential tilting (ET) estimator of Kitamura and
Stutzer (1997) and Imbens, Spady, and Johnson (1998). As shown by Smith
(1997), EL and ET share a common structure, being members of a class of
generalized empirical likelihood (GEL) estimators. We show that the CUE is
also a member of this class as are estimators from the Cressie and Read (1984)
power divergence family of discrepancies. All of these estimators and GMM
have the same asymptotic distribution but different higher order asymptotic properties.
We use the GEL structure, which helps simplify calculations and
comparisons, to analyze higher order properties like those of Nagar (1959).
We derive and compare the (higher order) asymptotic bias for all of these estimators.
We also derive bias corrected GMM and GEL estimators and consider
their higher order efficiency.