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

form. It can also take account of disturbance/error autocorrelation more or less
strictly specified, by reformulating the orthogonality conditions in an appropriate
way, as will be examplified below. This flexibility with respect to the imposition
of restrictions on the second order moments of disturbances/errors is one of the
primary virtues of GMM as compared with classical 2SLS. To operationalize the
latter method in the presence of unknown heteroskedasticity, we then first con-
struct consistent residuals bj , usually from (23), which we consider as a first step
GMM estimator, and estimate W_n by W^d_n = [n^{-2 P}_j z_j⁰b_j²z_j]^-1 ; see White (1984,
sections IV.3 and VI.2). Inserting this into (22) gives

(24)       βeGMM = [(^Pj x⁰_jz_j)(^Pj z⁰_jb_j²z_j)^-1(^Pj z⁰_jx_j)]^-1

× [(^Pj x⁰_jz_j)(^Pj z⁰_jb_j²z_j)^-1(^Pj z⁰_jy_j)].

The latter, second step GMM estimator, is in a sense an optimal GMM estimator
in the presence of unspecified error/disturbance heteroskedasticity. Both will be
considered in our empirical application below.

5 Simple GMM estimators

combining differences and levels

As explained in Section 4, the orthogonality conditions (OC’s) derived from eco-
nomic theory, (19), their empirical counterparts (20), and other restrictions im-
posed on second order moments of observed variables and errors and disturbances
play an essential role in GMM procedures. We have already made Assumption (A).
Before presenting the specific estimators for our panel data measurement error sit-
uation, we state the additional assumptions we will need.

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