2. GARCH(p, q) Processes

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

Published Online: 14 JUL 2010

DOI: 10.1002/9780470670057.ch2

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) GARCH(p, q) Processes, in GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470670057.ch2

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



  • conditional variance;
  • forecasting issues;
  • generalized autoregressive conditionally heteroscedastic (GARCH) models;
  • second-order stationarity conditions;
  • squaredreturn autocorrelations


In autoregressive conditionally heteroscedastic (ARCH) and their GARCH (generalized ARCH) models, the key concept is the conditional variance. In the classical GARCH models, the conditional variance is expressed as a linear function of the squared past values of the series. The ‘linear’ structure of these models can be displayed through several representations that are studied in this chapter. The chapter presents definitions and representations of GARCH models. Then it establishes the strict and second-order stationarity conditions. Starting with the first-order GARCH model, for which the proofs are easier and the results are more explicit, the chapter extends the study to the general case. It also studies the so-called ARCH(∞) models, which allow for a slower decay of squaredreturn autocorrelations. Then, the chapter considers the existence of moments and the properties of the autocorrelation structure. It concludes by examining forecasting issues.

Controlled Vocabulary Terms

autocorrelation; conditional variance; forecasting; generalized autoregressive conditional heteroskedasticity