8. Tests Based on the Likelihood

  1. Christian Francq1 and
  2. Jean-Michel Zakoïan1,2

Published Online: 14 JUL 2010

DOI: 10.1002/9780470670057.ch8

GARCH Models: Structure, Statistical Inference and Financial Applications

GARCH Models: Structure, Statistical Inference and Financial Applications

How to Cite

Francq, C. and Zakoïan, J.-M. (2010) Tests Based on the Likelihood, in GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470670057.ch8

Author Information

  1. 1

    University Lille 3, Lille, France

  2. 2

    CREST, Paris, France

Publication History

  1. Published Online: 14 JUL 2010
  2. Published Print: 23 JUL 2010

ISBN Information

Print ISBN: 9780470683910

Online ISBN: 9780470670057



  • asymptotic distribution;
  • GARCH(1, 1) model;
  • Portmanteau Tests;
  • qausi-maximum likelihood estimator (QMLE)


This chapter aims 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. It establishes the asymptotic distribution of the quasi-maximum likelihood estimator (QMLE) without assumption A5, at the cost of a moment assumption on the observed process. The asymptotic distribution obtained for the QMLE leads to modification of the standard critical regions. Two cases of particular interest are 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. The chapter devotes testing the adequacy of a particular GARCH (p, q) model, using portmanteau tests. It also contains a numerical application in which the preeminence of the GARCH(1, 1) model is questioned.

Controlled Vocabulary Terms

asymptotic distribution; generalized autoregressive conditional heteroskedasticity