Estimating GARCH by QML

1. Conditional Quasi-Likelihood
1. Asymptotic Properties of the QML Estimator
2. The ARCH(1) Case
3. The Non Stationary Case
2. ARMA-GARCH QML Estimation
3. Application to Real Data
4. Proofs of the Asymptotic Results*
5. Bibliographical Notes
6. Exercises

Keywords: ARMA-GARCH estimation, CLT (Central Limit Theorem) for Stationary Martingale Increments, Consistency and Asymptotic Normality, Ergodicity, Lyapounov's coefficient, GARCH estimation, Ordinary Least Squares (OLS) for estimating GARCH(p,q) models, Quasi Likelihood, Quasi-Maximum Likelihood (QML), Top Lyapounov exponent.

Description: The quasi-maximum likelihood 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 without any moment assumption on the observed process. By contrast, the least squares methods of the previous chapter require moments of order 4 at least. In this chapter, we study in details the conditional quasi-maximum likelihood method (conditional to initial values). We first consider the case when the observed process is a pure GARCH. We present an iterative procedure for computing the Gaussian log-likelihood, conditionally to fixed or random initial values. The likelihood is written as if the law of the variables η_t was Gaussian N(0, 1) (we talk about pseudo or quasi-likelihood), but this assumption is not necessary for the strong consistency of the estimator. In a second time, we will study the application of the method to the estimation of ARMA-GARCH models. The asymptotic properties of the quasi-maximum likelihood (QML) estimator are established at the end of the chapter.

R Program for Figure 2

Programme R for Table 7.4 (GARCH(1,1) estimation of 11 stock market returns)

Data used in Section 7.3 daily returns of 11 indices