Estimating GARCH Models: When to Use What?

The class of GARCH models has proved particularly valuable in modelling time series with time varying volatility. These include financial data, which can be particularly heavy tailed. It is well understood now that the tail heaviness of the innovation distribution plays an important role in determining the relative performance of the two competing estimation methods, namely the maximum quasilikelihood estimator based on a Gaussian likelihood (GMLE) and the logtransform based least absolutely deviations estimator (LADE); see Peng and Yao (2003). A practically relevant question is when to use what. We provide in this paper a solution to this question. By interpreting the LADE as a version of the maximum quasilikelihood estimator under the likelihood derived from assuming hypothetically that the log squared innovations obey a Laplace distribution, we outline a selection procedure based on some goodness-of-fit type statistics. The methods are illustrated with both simulated and real data sets. Although we deal with the estimation for GARCH models only, the basic idea may be applied to address the estimation procedure selection problem in a general regression setting. Some key words: Estimation procedure selection; GARCH; Gaussian likelihood; Heavy tail; Laplace distribution; Least absolute deviations estimator; Maximum quasilikelihood estimator; Time series.