GARCH(1,1) reaction to one-off shocks

The GARCH approach models the variance using the prediction errors \(e_t\) (also called shocks or unexpected returns). The parameter \(\alpha\) determines the reactivity to \(e_t^2\) , while \(\beta\) is the weight on the previous variance prediction.

In this exercise, we consider the series of squared prediction errors e2 <- c(10,25,rep(10,20)). We plot the variance for:

\(\alpha=0.1\) and \(\beta=0.8\)
\(\alpha=0.19\) and \(\beta=0.8\)
\(\alpha=0.1\) and \(\beta=0.89\).

We set \(\omega\) such that the long term variance is 10.

Which statement about the effect of the shock on the variance is wrong?

Possible answers

The variance shoots up and then returns to \(\omega/(1-\alpha-\beta)\).

The larger is \(\alpha\), the larger is the immediate impact of the shock.

Holding fixed \(\alpha\), the larger is \(\beta\), the longer is the duration of the impact.

The triangles show the variance when \(\alpha=0.1\) and \(\beta=0.8\).

Exercise

GARCH(1,1) reaction to one-off shocks

Instructions

Possible answers