A GARCH model is estimated and a maximised LLF of 66.85 is obtained. Suppose that a researcher wishes to test whether β = 0 in

(9.77)

(9.76)

(9.77)

The model is estimated imposing the restriction and the maximised LLF falls to 64.54. Is the restriction supported by the data, which would correspond to the situation where an ARCH(1) specification was sufficient? The test statistic is given by

(9.78)

The test follows a χ

2

(1) = 3.84 at 5%, so that the null is marginally rejected. It would thus be concluded that an ARCH(1) model, with no lag of the conditional variance in the variance equation, is not quite sufficient to describe the dependence in volatility over time.