Handling the measurement error problem by means of panel data: Moment methods applied on firm data

1,...,n, and define the vector valued (G × 1) function of corresponding means taken
over all available observations,

(20)                 gn(y^, x^, z;^β) = (1 /n) ∑n=ι^zj(y_j ^{- χ}j^β)∙

It may be considered the empirical counterpart to E[z⁰(y - xβ)] based on the
sample. The essence of GMM is to choose as an estimator for β the value which
brings the value of g_n (y, x, z; β) as close to its theoretical counterpart, the zero
vector 0G,1, as possible. If G = K, an exact solution to the equation g_n (y, x, z; β)=
0G,1 exists and is the simple IV estimator

(21)                            β* = [P _j z'jXj ] - ¹[P _j zjy_i ] ∙

If G>K, which is the most common situation, the GMM procedure solves the
estimation problem by minimizing a distance measure represented by a quadratic
form in gn(y, x, z; β) for a suitably chosen positive definit (G × G) weighting matrix
W_n, i.e.,

(²²)   ^βGMM = ^βGMM(Wn) = argminβ^[gn(y^, x^{, z};^β)⁰W_ng_n(y^{, χ,} z;^β)^]∙

All estimators obtained in this way are consistent. The choice of W_n determines
the efficiency of the method. A choice which leads to an asymptotically efficient
estimator of β, is to set this weighting matrix equal (or proportional) to the inverse
of (an estimate of) the (asymptotic) covariance matrix of (1/n) ^P_jⁿ₌₁ z_j⁰_j; see, e.g.,
Davidson and MacKinnon (1993, Theorem 17.3) and Harris and Matyas (1999,
section 1.3.3).

If is serially uncorrelated and homoskedastic, with variance σ² , the appropriate
choice is simply W_n = [n^-2σ^{2 P}_jⁿ₌₁ z_j⁰z_j]^-1. The resulting estimator obtained from
(22) is

(23)β^bGMM =[(^Pjx⁰_jz_j)(^Pjz⁰_jz_j)^-1(^Pjz⁰_jx_j)]^-1[(^Pjx⁰_jz_j)(^Pjz⁰_jz_j)^-1(^Pjz⁰_jy_j)],

which is the standard Two-Stage Least Squares (2SLS) estimator. The method can
also be fruitfully applied if j has a heteroskedasticity of unspecified (or unknown)

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