Advanced Econometric Methods by Thomas B. Fomby, R. Carter Hill, Stanley R. Johnson PDF

By Thomas B. Fomby, R. Carter Hill, Stanley R. Johnson

ISBN-10: 0387968687

ISBN-13: 9780387968681

ISBN-10: 1441987460

ISBN-13: 9781441987464

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Example text

O (X'O-l X /T) = Qn is finite and nonsingular. The structure now assumed for the errors permits the possibility that they may be correlated and/or may have different variances. In either case, knowledge of the properties of the errors should be used in the estimation process. This conclusion is formalized by Aitken's theorem, Aitken (1935). 1 (Aitken's Theorem). In the generalized least squares model the estimator p= (X'O-l X)-l X'O -ly is efficient among the class of linear, unbiased estimators of p.

The ijth element of the information matrix is [ .. = _E[02 In L(x; 9)J IJ 00; oOj , i, j = 1, 2, .. , K. Let ()be any unbiased estimatorof9 with covariance matrix L . Then the matrix L - [l(9)J -1 is positive semidefinite. 7). The remarkable result of the Cramer-Rae theorem is that the lower bound for the variance at any unbiased estimator of the parameters of a 34 3 Point Estimation and Tests of Hypotheses in Small Samples regular density is [I(9)r 1, regardless of the estimation technique used to derive it.

The expectation of the derivative of the natural logarithm ofthe likelihood function of a random sample from a regular density is zero. 3) with respect to 0 provides 02 In L(x; 0) L( . 0) d 002 x, X J = + JO In L(x ; 0) oL(x; 0) d 00 00 x 02 In L(x ; 0) 002 L(x ; 0) dx J + J[o In L(x; O)J2 00 L(x ; 0) dx = O.