Advanced Econometrics Takeshi Amemiya
Two Error Components Model with a Serially Correlated Error
In the subsection we shall discuss the 2ECM defined by Eqs. (6.6.18) and
(6.6.19) in which e follows an AR(1) process, that is,
in = УЧ,-1 + 4,
where {<!;„} are i. i.d. with zero mean and variance cj. As in the Balestra-Ner - love model, the specification of єю will be important if T is small. Lillard and Willis (1978) used model (6.6.39) to explain the log of earnings by the independent variables such as race, education, and labor force experience. They assumed stationarity for (ей), which is equivalent to assuming Еею = 0, Уєю = a/{ 1 — у2), and the independence of єю from £(1, £a,. . . . Thus, in the Lillard-Willis model, Euu' = (a2^TVT+Y) ® L,, where Г is like (5.2.9) wither2 = cr^andp = y. Lillard and Willis estimated fi and p by LS and <r2, a, and у by inserting the LS residuals into the formulae for the normal MLE. Anderson and Hsiao (1982) have presented the properties of the full MLE in this model.
The model of Lillard and Weiss (1979) is essentially the same as model
(6.6.39) except that in the Lillard-Weiss model there is an additional error component, so that u = Lp + [I — L(L'L)-1L'](I ® f)f + e, where f = (1,2,. . . , T)' and {is independent of/i and e. The authors used LS, FGLS, and MLE to estimate the parameters of their model.10 The MLE was calculated using the LISREL program developed by Joreskog and Sorbom (1976). Hause (1980) has presented a similar model.
Finally, MaCurdy (1982) generalized the Lillard-Willis model to a more general time series process for eit. He eliminated nt by first differencing and treating уu — Уц— і as the dependent variable. Then he tried to model the LS predictor for e„ — €/',-1 by a standard Box-Jenkins-type procedure. MaCurdy argued that in a typical panel data model with small T and large N the assumption of stationary is unnecessary, and he assumed that the initial values єю, . . . are i. i.d. random variables across і with zero mean and unknown variances.
