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  1. Tests Based on the Likelihood

    1. Test of the Second Order Stationarity Assumption
    2. Asymptotic Distribution of the QML when θ_0 Is at the Boundary
      1. Computation of the Asymptotic Distribution
    3. Significance of the GARCH Coefficients
      1. Presentation of the Main Tests
      2. Modification of the Standard Tests
      3. Test for the Nullity of One Coefficient
      4. Conditional Homoscedasticity Tests with ARCH Models
      5. Asymptotic Comparison of the Tests
    4. Diagnostic Checking with Portmanteau Tests
    5. Application: Is the GARCH(1,1) Model Over-Represented?
    6. Proof of the Main Results*
    7. Bibliographical Notes
    8. Exercises
Christian Francq and Jean-Michel Zakoïan

Keywords: ARCH Effect Test, Asymptotic Law of the QML at the Boundary, Bahadur's Approach, QML for GARCH, Likelihood Ratio, Pitman's Approach, Score Test Test, Test of Student, Wald Test.

Description: In the previous chapter, we have seen that the asymptotic normality of the QML estimator 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 constitutes the object of the first section. 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 the positivity of the estimated conditional variance. Without these constraints, it is impossible to compute the Gaussian log-likelihood of the GARCH model. Moreover, asymptotic normality of the QML has been established assuming that the parameter belongs to the interior of the parameter space (assumption A5). 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 the QMLE can not be Gaussian. Before considering the significance tests, we shall thus establish in Section 8.2 the asymptotic distribution of the QML estimator without Assumption A5, at the price of a moment assumption on the observed process. In Section 8.3, we present the main tests (Wald, score and likelihood ratio) used for testing the nullity of some coefficients. The asymptotic distribution obtained for the QML estimator will lead to modify the standard critical regions. Two cases of particular interest will be examined in detail: the test of nullity of only one coefficient and the test of conditional homoscedasticity, which corresponds to the nullity of all the coefficients αi and βj. Section 8.4 is devoted to testing the adequacy of a particular GARCH(p,q) model, using portmanteau tests. The chapter also contains a numerical application in which the preeminence of the GARCH(1,1) is questioned.

Programmes R utilisés pour le Tableau 8.1 R tests l'hypothèse de variance infinie

Programmes R utilisés pour le Tableau 8.4 R tests du modèle GARCH(1,1) pour les rendements journaliers , R pour les rendements hebdomadaires , R pour les taux de change

Données utilisées dans la partie 8.1 et la partie 8.4 données rendements journaliers , données rendements hebdomadaires ,données taux de change (obtenues sur yahoo).