 Solution

The score equation $X^T(\mu -Y)=0$ is independent of $\gamma$, because $\gamma >0$ is only a common factor. Fixing $\varphi =1$ or estimating it therefore gives the same maximum-likelihood estimator $\hat{\beta }$.

Under the usual full-rank and regularity conditions,with $W$ evaluated consistently at the fitted means. Consequently every standard error from "mod2" is $\sqrt{\hat{\varphi }}$ times the corresponding standard error computed with dispersion one in "mod1". In particular,