Advanced Econometrics Takeshi Amemiya
A Useful Transformation for the Calculation of the Feasible Generalized Least Squares Estimator
Let Rt be the matrix obtained by inserting p into the right-hand side of
(5.2.10) . Then we have by (5.2.14) fip = (X'ft'&X^X'RJRj. Thus fip can be obtained by least squares after transforming all the variables by Rt. Another, somewhat simpler transformation is defined by eliminating the first row of R,. This is called the Cochrane-Orcutt transformation (Cochrane and
Orcutt, 1949). The resulting estimator is slightly different from FGLS but has the same asymptotic distribution.
If {u,} follow AR(2) defined by (5.2.16), the relevant transformation is given by (5.2.27), where a’s are determined by solving V(axux) = a2, У(а2щ + a3u2) = a2, and Е[ахщ{а2щ + a3u2)] = 0. A generalization of the Cochrane-Orcutt transformation is obtained by eliminating the first two rows of R2. Higher-order autoregressive processes can be similarly handled.
