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The quasi-likelihood function optimized under GEE is:

$S_k(\beta)=\sum_{i=1}^{K}\frac{\partial\mu_i}{\partial\beta_k}\nu_i^{-1}(y_i-\mu_i)=0,$

where $\mu_i=h(\textbf{x}_i,\beta)$ is the conditional mean of $y_i$ and $\nu_i$ is the conditional variance of $y_i$.

I know I'm wrong here, but my thought process is that, since the quasi-likelihood function incorporiates the variance, and it is not separable from the mean (and hence the betas), that different assumptions about the variance structure should influence all the parameter estimates, including the betas.

By somewhat of an analogy (again, I'm sure this analogy is inappropriate but I'm not entirely clear why), FGLS estimation builds assumptions about the covariance matrix into its optimization routine and yields different parameters from OLS because of it.

So my question is, given that the quasi-likelihood function is a function of both the conditional mean and the conditional variance, and the two terms interact in a multiplicative fashion, why is it that the beta estimates do not depend on the form of the variance?

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  • $\begingroup$ Your quasi-likelihood function is written incorrectly; the $(y_i-\mu_i)$ term should be squared. $\endgroup$
    – Donnie
    Commented Jul 4, 2015 at 21:27
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    $\begingroup$ All the sources I've encountered, including the original Zeger and Liang articles, have $(y_i-\mu_i)$ as a first-order term. $\endgroup$
    – Yakkanomica
    Commented Jul 5, 2015 at 3:59

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