I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation.
$$ -\operatorname{E}_{t-1} \left[ \frac{\partial^2\ln p(y_t\mid f_t;\theta)}{\partial\sigma_t^2\,\partial\sigma_t^2} \right]^{-1} = \frac{2\sigma_t^4 (\nu+3)} \nu. $$
After considering the log likelihood, the score is obtained by differentiation. I'm having troubles in two points:
1) Showing that the expectation of the score equals zero, in this specific case;
2) How do I get the negative of the inverse of the information Matrix as it is presented? I tried to differentiate another time the score w.r.t. $\sigma^2$ and then taking the expectation, but I don't know how to get the given result... Obviously I think that this problem is related to my inability to show the zero expectation of the score.
Thanks to anybody that wants to help!