Nonspherical Disturbances - The Generalized Regression Model

This and the next two exercises are based on the test statistie usually used to test a set of J linear reshietions in the generalized regression model  

use the inverse square root mahix of Q, P = to obtain the transformed data. 

We now turn to the denominator. By multiplying it out, we find that the denominator is (y - X P) (y - X P)/(n-/Q. This is exactly the sum of squared residuals in the least squares regression of y on X. Since y = X p + e and P = (X X ) X y the denominator is e M e / n - K), the familiar form of the sum of squares. Once again, this is an idempotent quadratic form in a normal vector (and, again, apart from the scale factor, which now cancels). The rank of the M matrix is n - X, as always, so the denominator is also a chi-squared variable divided by its degrees of freedom. 

It remains only to show that the two chi-squared variables are independent. We know they are if the two matiices are orthogonal. They are since MX =0. This completes the proof, since all of the requfrements for the F distribution have been shown. 

we know that the denominator of the F statistic converges to Therefore, the limiting disfribvrtion of the F statistic is the same as the limiting distribution of the statistic which results when the denominator is replaced by It is usefrrl to write this modified statistic as W = (l/CT )(Rp -q) [R(X X )- R ]- (Rp-q)//. 

incorporate the results from the previous problem to write this as 

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