Two Error Components Model with Endogenous Regressors
6.6.5 The two error components model with endogenous regressors is defined by У! = Xifi + Zy + YlS + n + el (6.6.40)
У2 = X2)? + Zy + + fl + €2 у 7* = Xrfi T Zy + Yr£ + Ц + eT,
where y„ t = 1, 2,. . . , T, is an А-vector, X, is an NX К matrix of known constants, Z is an NX G matrix of endogenous variables, Y, is an iV X F matrix of endogenous variables, and ц and e, are W-vectors of random variables with the same characteristics as in 2ECM. The variable Z is always assumed to be correlated with bothp and e,, whereas Y, is sometimes assumed to be correlated with both fi and є, and sometimes only with fi. The two error components model with endogenous regressors was analyzed by Chamberlain and Griliches (1975) and Hausman and Taylor (1981). Chamberlain and Griliches discussed maximum likelihood estimation assuming normality, whereas Hausman and Taylor considered the application of instrumental variable procedures.
Amemiya and MaCurdy (1983) proposed two instrumental variables estimators: one is optimal if Y, is correlated with e, and the other is optimal if Y, is uncorrelated with When we write model (6.6.40) simply as у = Wa + u, the first estimator is defined by
a, = (W’Q“1’2P1n-1/2W)“,W/£|-1/aPlQ-,/ay, (6.6.41)
where P, is the projection matrix onto the space spanned by the column vectors of the NTXKT2 matrix fl_1/2(Ir© S), where S = (X,, X2,. . . , XT). Amemiya and MaCurdy have shown that it is asymptotically optimal among all the instrumental variables estimators if Y, is correlated with €,. The second estimator is defined by
where P2 = I — Г_11г1г® [Ijv S(S’S)-1S’]. It is asymptotically optimal among all the instrumental variables estimators if Yt is uncorrelated with et. The second estimator is a modification of the one proposed by Hausman and Taylor (1981). In both of these estimators, Cl must be estimated. If a standard consistent estimator is used, however, the asymptotic distribution is not affected.