I know that the calculation of parameter values of a standard OLS can be made more efficient using a QR decomposition;

i.e. if $X=QR$ and we are using the model $Y=X\beta+\epsilon$;

Then it is true that $R\beta=Q^TY$ and therefore we can make the computation of $\hat{\beta}$ more efficient.

My question is, is there an equivalent QR decomposition trick for the generalised least squares estimator:



1 Answer 1


If by QR decomposition trick you mean "can one use a QR decomposition to fit a weighted least squares?", then the answer is yes, provided one knows the value of $W$ and $W$ is positive definite.

If $W$ is positive definite then you can take the Cholesky decomposition of $W$, call it $chol(W)$ so that $chol(W)^Tchol(W)=W$. Then you can write $chol(W)X=\tilde{X}$ and $chol(W)Y = \tilde{Y}$. Then take the QR decomposition of $chol(W)X$ and you should have your result.

In practice the positive definite restriction on $W$ is not too restrictive. Generalized linear models for example have $W$ a diagonal matrix of weights calculated from previous iteration of weighted least squares solution. The weights are all non-negative by construction so this obviates or at least sidesteps the p.d. issue.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.