Homework 6

Generalized Empirical likelihood

(Due 4/2)

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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, the continuous updating estimator (CUE) and the exponential tilting (ET) estimator. All of these estimators and GMM have the same asymptotic distribution but different higher-order asymptotic properties.

EL has two theoretical advantages. First, its asymptotic bias does not grow with the number of moment restrictions, while the bias of the others often does. Consequently, for large numbers of moment conditions the bias of EL will be less than the bias of the other estimators. This property is important in econometrics, where many moment conditions are often used. For example, Hansen and Singleton (1982) and Arellano and Bond (1991) all use quite large numbers of moment conditions. The relatively low asymptotic bias of EL indicates that it is an important alternative to GMM in such applications. The second theoretical advantage of EL is that after it is bias corrected, using probabilities obtained from EL, it is higher efficient relative to the other estimators. This property has a simple explanation. When the data are discrete, having finite support, EL is equal to the MLE (can you show it?). Consequently, for discrete data EL inherits the well known higher order efficiency of MLE. Then, because discrete distributions can be used to approximate moments of a continuous distribution the efficiency of EL for the discrete case leads to efficiency in general.

Let be iid observations on a data vector z. Also let be a parameter vector and g be an vector of functions of the data observation and the parameter, where The model has a true parameter satisfying the moment condition

where denotes expectation taken with respect to the distribution of An important estimator of is the two-step GMM estimator of Hansen (1982). To describe it, let



and

Also, let be some preliminary estimator, given by

where is a random matrix with properties to be specified below. The GMM estimator is

The alternatives to GMM we consider are generalized empirical likelihood (GEL) estimators. To describe GEL let be a function of a scalar that is concave on its domain, an open interval containing zero. The estimator is the solution to a saddle point problem

The EL estimator is a special case with

and The exponential titling estimator is a special case with

The CUE is analogous to GMM except that the objective function is simultaneously minimized over in It is given by

where denotes any generalized inverse of a matrix , satisfying

The following results shows that if is quadratic then Let

and be an n-vector of units. Thus,

and

By Rao (1973), is invariant to the generalized inverse (ginv) and for any ginv. Then the CUE objective function

is invariant to ginv.

1. Show that if is quadratic, then the second-order Taylor expansion of in about zero is exact where That is
   
   

By concavity of in any solution to the first-order conditions

will maximize with respect to holding fixed. Then

so that

solves the first-order conditions. Since

the GEL objective function is a monotonic increasing transformation of the CUE objective function,so that the set of GEL estimators concides with the set of CUE estimator.

Consider the following linear model, where the structure equation is given by

and the reduced form for Y by

where is and is Assume the order condition for identification.

Under strong identification of , is fixed matrix of full column rank. Weak identification (Staiger and Stock, 1997) is modeled by letting the correlation between the instruments and the endogenous variables fade away as n goes to infinity. Assume

where is a fixed matrix. For given sample size n, define the random -vector

Define the matrix

Let

Assume (i) are iid, (ii) (iii) Next, we give a formal definition of the GEL estimator, of It exploits the moment condition

and is given as the solution to a saddle point problem

where

For quadratic a second order Taylor expansion of in about zero is exact. This implies that for each the maximization in in () is unconstrained. It follows that for

and



It holds that

By concavity of in any solution to the FOC

maximizes with respect to for fixed Then

The next lemma establishes the limit process of under weak identification.

The theorem shows that under weak identification the GEL estimator has a nonstandard distribution and is in general inconsistent.