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We have the following model (GARCH (1,1) )

$y_t=\sigma_t\epsilon_t$

$\sigma_t^2 = \omega + \beta*\sigma_{t-1}^2 + \gamma*y_{t-1}^2$

Note that we can rewrite the latter as: $\sigma_t^2=\frac{\omega}{1-\beta} +\gamma\sum_{j=1}^\infty \beta^{j-1}y_{t-j}^2$

Can anybody show the algebra of the impulse response function of $\sigma_t^2$ with respect to $\epsilon_{t-k}^2$?

Many thanks!!

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