Test of Hypotheses
In order to test H0; Ев = r, under the general variance-covariance matrix assumption, one can revert to the transformed model (9.3) which has a scalar identity variance-covariance matrix and use the test statistic derived in Chapter 7
(RPgls — r)'[E(X*X*)-1E>]-1(RPgls — r)/a2 – x2 (9.15)
Note that вGLS replaces вOLS and X* replaces X. Replacing X* by P 1X, we get (RPgls — r)'[R(X’Q-1 X)-1R’]-1(r3gls — r)/a2 – x2
This differs from its counterpart in the spherical disturbances model in two ways. вGLS replaces вoLS, and (X’Q-1X) takes the place of X’X. One can also derive the restricted estimator based on the transformed model by simply replacing X* by P-1X and the OLS estimator of в by its GLS counterpart. Problem 4 asks the reader to verify that the restricted GLS estimator is
eRGLS = eGLS – (X’Q-1X)-1R'[R(X’Q-1X)-1 R’]-1(RdGLS — r) (9.17)
Furthermore, using the same analysis given in Chapter 7, one can show that (9.15) is in fact the Likelihood Ratio statistic and is equal to the Wald and Lagrangian Multiplier statistics, see Buse (1982). In order to operationalize these tests, we replace a2 by its unbiased estimate s*2, and divide by g the number of restrictions. The resulting statistic is an F(g, n — K) for the same reasons given in Chapter 7.