# 7. Estimating GARCH Models by Quasi-Maximum Likelihood

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

Published Online: 14 JUL 2010

DOI: 10.1002/9780470670057.ch7

## GARCH Models: Structure, Statistical Inference and Financial Applications

#### How to Cite

Francq, C. and Zakoïan, J.-M. (2010) Estimating GARCH Models by Quasi-Maximum Likelihood, in GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons, Ltd, Chichester, UK. doi: 10.1002/9780470670057.ch7

#### 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

## SEARCH

### Keywords:

• ARMA-GARCH Models;
• asymptotic properties;
• GARCH models;
• Gaussian log-likelihood;
• quasi-maximum likelihood estimator (QMLE)

### Summary

The quasi-maximum likelihood (QML) method is particularly relevant for GARCH models because it provides consistent and asymptotically normal estimators for strictly stationary GARCH processes under mild regularity conditions, but with no moment assumptions on the observed process. This chapter details the conditional QML method and first considers the case when the observed process is pure GARCH. It presents an iterative procedure for computing the Gaussian log-likelihood, conditionally on fixed or random initial values. The chapter focuses on the application of the method to the estimation of ARMA-GARCH models. It establishes the asymptotic properties of the quasi-maximum likelihood estimator (QMLE). The chapter employs the QML method to estimate GARCH(1, 1) models on daily returns of 11 stock market indices, namely the CAC, DAX, DJA, DJI, DJT, DJU, FTSE, Nasdaq, Nikkei, SMI and S&P 500 indices.

#### Controlled Vocabulary Terms

asymptotic distribution; consistent estimator; Gaussian process; generalized autoregressive conditional heteroskedasticity