3.16 Kurtosis of GARCH Models
Uncertainty in volatility estimation is an important issue, but it is often overlooked. To assess the variability of an estimated volatility, one must consider the kurtosis of a volatility model. In this section, we derive the excess kurtosis of a GARCH(1,1) model. The same idea applies to other GARCH models, however. The model considered is
where α0 > 0, α1 ≥ 0, β1 ≥ 0, α1 + β1 < 1, and {ϵt} is an iid sequence satisfying
where Kϵ is the excess kurtosis of the innovation ϵt. Based on the assumption, we have the following:
- Var(at) = .
- provided that exists.
Taking the square of the volatility model, we have
Taking expectation of the equation and using the two properties mentioned earlier, we obtain
provided that 1 > α1 + β1 ≥ 0 and The excess kurtosis of ...
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