By Fallah F
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N. The ML estimator of β is consistent but that of σ2 is not. 2 Conditional Likelihood In the linear static model, does not depend on ηi is a sufﬁcient statistic for ηi. 49) we obtain: which does not depend on ηi because it is only a function of the within-group errors. 55) is a function of β and σ2 which can be used as an alternative basis for inference. 25). 3 Marginal (or Integrated) Likelihood Finally, we may consider the marginal distribution of yi given xi but not ηi: where F (ηi | xi) denotes the conditional cdf of ηi given xi.
On other occasions, correlation between regressors and individual effects can be regarded as an empirical issue. In these cases testing for correlated unobserved heterogeneity can be a useful speciﬁcation test for regression models estimated in levels. Researchers may have a preference for models in levels because estimates in levels are in general more precise than estimates in deviations (dramatically so when the time series variation in the regressors relative to the cross-sectional variation is small), or because of an interest in regressors that lack time series variation.
69) Feasible approaches to optimal estimation can be based on an estimator of B(zi). 40) with Zi = (IT − 1 ⊗ zi′). 71) and are asymptotically equivalent. On the other hand, if If Ω(zi) = σ2IT−1 and E(Xi | zi) is linear in zi, then zit = xit the statistic boils down to the ordinary within-group estimator. 3 Nonlinear Simultaneous Equations Finally, we consider a system of g nonlinear simultaneous equations with additive effects. The previous models can be regarded as special cases of this one with g = 1.