Variance as an Exponential Function of Regressors
As we mentioned before, the linear specification, however simple, is more general than it appears. However, a researcher may explicitly specify the variance to be a certain nonlinear function of the regressors. The most natural choice is an exponential function because, unlike a linear specification, it has the attractive feature of ensuring positive variances. Harvey (1976), who assumed yt — exp (z’,a), proposed estimating a by LS applied to the
regression of log fi? on z, and showed that the estimator is consistent if 1.2704 is subtracted from the estimate of the constant term. Furthermore, the estimator has an asymptotic covariance matrix equal to 4.9348(Z’Z)-‘, more than double the asymptotic covariance matrix of MLE, which is 2(Z’Z)~I.
Error components models are frequently used by econometricians in analyzing panel data—observations from a cross-section of the population (such as consumers, firms, and states, and hereafter referred to as individuals) at various points in time. Depending on whether the error term consists of three components (a cross-section-specific component, a time-specific component, and a component that depends both on individuals and on time) or two components (excluding the time-specific component), these models are called three error components models (3ECM) and two error components models (2ECM), respectively. Two error components models are more frequently used in econometrics than 3ECM because it is usually more important to include a cross-section-specific component than a time-specific component.
For example, in studies explaining earnings, to which error components models have often been applied, the cross-section-specific component may be regarded as the error term arising from permanent earnings and is more important than the time-specific component.
We shall discuss first 3ECM and then various generalizations of 2ECM. These generalizations are (1) the Balestra-Nerlove model (2ECM with lagged endogenous variables), (2) 2ECM with a serially correlated error, and (3) 2ECM with endogenous regressors.