8
Tests Based on the Likelihood
In the previous chapter, we saw that the asymptotic normality of the QMLE of a GARCH model holds true under general conditions, in particular without any moment assumption on the observed process. An important application of this result concerns testing problems. In particular, we are able to test the IGARCH assumption, or more generally a given GARCH model with infinite variance. This problem is the subject of Section 8.1.
The main aim of this chapter is to derive tests for the nullity of coefficients. These tests are complex in the GARCH case, because of the constraints that are Imposed on the estimates of the coefficients to guarantee that the estimated conditional variance is positive. Without these constraints, it is impossible to compute the Gaussian log-likelihood of the GARCH model. Moreover, asymptotic normality of the QMLE has been established assuming that the parameter belongs to the interior of the parameter space (assumption A5 in Chapter 7). When some coefficients αi or βj are null, Theorem 7.2 does not apply. It is easy to see that, in such a situation, the asymptotic distribution of 
(
n − θ0) cannot be Gaussian. Indeed, the components in of n are constrained to be positive or null. If, for instance, θ0i = 0 then (in − θ0i) = in ≥ 0 for ...
