Recursive Estimation of GARCH Models

Abstract

This article develops three recursive on-line algorithms, based on a two-stage least squares scheme for estimating generalized autoregressive conditionally heteroskedastic (GARCH) models. The first one, denoted by 2S-RLS, is an adaptation of the recursive least squares method for estimating autoregressive conditionally heteroskedastic (ARCH) models. The second and the third ones (denoted, respectively, by 2S-PLR and 2S-RML) are adapted versions of the pseudolinear regression (PLR) and the recursive maximum likelihood (RML) methods to the GARCH case. We show that the proposed algorithms give consistent estimators and that the 2S-RLS and the 2S-RML estimators are asymptotically Gaussian. These methods seem very adequate for modeling the sequential feature of financial time series, which are observed on a high-frequency basis. The performance of these algorithms is shown via a simulation study.

Keywords:

• GARCH process
• Martingale convergence theorem
• On-line esti-mation
• Two-stage pseudolinear regression
• Two-stage recursive least squares

Mathematics Subject Classification:

• 62F12
• 62M10

Acknowledgments

The authors express their most sincere thanks and grateful acknowledgment to the editor N. Balakrishnan, an associate editor, and Professor Mohamed Bentarzi for their helpful comments and useful suggestions.