# Do you know how to compute the IRF of a GARCH (1,1)

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!!