A. Colin Cameron 735There are several attractions to GMM. First, it provides a unifying framework to
estimation as it nests many estimation procedures, including LS, withh(wi,θ)=
yi−x′iβ, and ML, withh(wi,θ)=∂lnf(yi|xi,θ)/∂θ, as special cases. Second, it provi-
des a natural extension of instrumental variables methods in overidentified models
from linear to nonlinear models, and can be viewed as a generalization of non-
linear 2SLS. Third, it views estimation as a sample analog to population moment
conditions, the analogy principle emphasized by Manski (1988). Fourth, taking
this view leads naturally to conditional moment tests (see section 14.4.2) that lead
to model moment specification tests based on model moment conditions that are
not exploited in estimation. Finally, it is relatively simple computationally as stan-
dard iterative methods such as Newton–Raphson can be employed, though not all
econometric packages provide a general GMM command for nonlinear models.
A method closely related to GMM, though one less used, is minimum distance
estimation. Suppose that the relationship betweenqstructural parameters andr>
qreduced form parameters is thatπ =g(θ). Given a consistent estimatêπof
the reduced form parameters, an obvious estimator iŝθsuch that̂π=g(̂θ). But
this is infeasible sinceq<r. Instead, the minimum distance (MD) estimator̂θMD
minimizes, with respect toθ, the objective function:
QN(θ)=(̂π−g(θ))′WN(̂π−g(θ)), (14.4)whereWNis anr×rweighting matrix. The optimal MD estimator uses the weight-
ing matrixWN=̂V[̂π]−^1 in (14.4). This estimator is used mainly in panel data
analysis (see Chamberlain, 1982, 1984), especially in estimation of covariance
structures (see Abowd and Card, 1987).
The statistics literature rarely uses the GMM framework. This may be because
GMM is particularly useful for overidentified models, notably IV with surplus
instruments, that are much more often used in econometrics. Instead, for non-
linear models the statistics literature emphasizes the more restrictive generalized
linear models and generalized estimating equations frameworks (see McCullagh
and Nelder, 1983).
14.3.2 Empirical likelihood
Empirical likelihood is based on the same moment conditions as GMM, but is
a different estimation method with better second-order asymptotic properties.
Empirical likelihood may be a better estimator in settings where optimum GMM
is known to perform poorly in finite samples, but it is not widely used, in part due
to computational difficulty.
Letπi=f(yi|xi)denote the probability that theith observation onytakes the
realized valueyi. The empirical likelihood (EL) approach, introduced by Owen
(1988), maximizes the empirical log-likelihood function:
QN(π 1 ,...,πN)=N−^1∑Ni= 1lnπi, (14.5)subject to any model constraints.