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working through Peter McCullagh's glm book and having a hard time with understanding quasi-likelihood. I'm working on this question below and I think I need to find the quasi-likelihood estimates and then use a wald or score test? I'm very lost.

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The question does not ask you to find the beta estimates or to conduct a test. The only thing you need is the asymptotic covariance matrix, which is given earlier in the same Chapter in the book. You only need to substitute into the covariance matrix the $X$ matrix, the link derivative and the variance function, all of which are specified as part of the question.

The point of this problem in McCullagh's book is that, in this particular setup with a log-link and a squared variance function, the link derivative and the variance function cancel out of the covariance matrix, leaving a very simple formula.

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