Dear list members
I have a WEIGHTED positive definite matrix.
I use Cholesky decomposition to take its square root.
Call this matrix L
The model is Y= Xb +e
Multiplying through gives LY=LXb +Le
or Y*=X*b +e*
solving gives (bhat)= (X*transpose X*)^-1 by X*transpose Y*
and var(dhat)= sigmasquared (X*transpose X*)^-1
The result of this gives (for the model above) a 2 x 2 var-cov matrix.
Now the usual thing is to get the variance of the column of residuals
ie. sigmasquared and multiply this value into the variance-covariance matrix
and the variance of b1 is the (1,1) value and the variance of the b2
estimator
is the (2,2) value.
BUT (here is the question, please) do you need to
multiply the variance of the residuals into the var-covar matrix when the
covariance matrix from whence L was computed is weighted, or not.
In usual GLS I think that you do. But what about when weighted var-covars
are used. I think that the variance of the resids is already incorporated
into the
(1,1) and (2,2) cell values, yes or no?
Thanks everybody, for your help. Richard Hudson Qld Austarlia
|