Chapter 49 ARCH MODELS a TIM BOLLERSLEV Northwestern University and N.B.E.R. ROBERT F. ENGLE University of California, San Diego and N.B.E.R. DANIEL B. NELSON University of Chicago and N.B.E.R. Contents Abstract 2961 1. Introduction 2961 1.1. Definitions 2961 1.2. Empirical regularities of asset returns 2963 1.3. Univariate parametric models 2967 1.4. ARCH in mean models 2972 1.5. Nonparametric and semiparametric methods 2972 2. Inference procedures 2974 2.1. Testing for ARCH 2974 2.2. Maximum likelihood methods 2977 2.3. Quasi-maximum likelihood methods 2983 2.4. Specification checks 2984 aThe authors would like to thank Torben G. Andersen, Patrick Billingsley, William A. Brock, Eric Ghysels, Lars P. Hansen, Andrew Harvey, Blake LeBaron, and Theo Nijman for helpful comments. Financial support from the National Science Foundation under grants SES-9022807 (Bollerslev), SES- 9122056 (Engle), and SES-9110131 and SES-9310683 (Nelson), and from the Center for Research in Security Prices (Nelson), is gratefully acknowledged. Inquiries regarding the data for the stock market empirical application should be addressed to Professor G. William Schwert, Graduate School of Management, University of Rochester, Rochester, NY 14627, USA. The GAUSS TM code used in the stock market empirical example is available from Inter-University Consortium for Political and Social Research (ICPSR), P.O. Box 1248, Ann Arbor, MI 48106, USA, telephone (313)763-5010; Order "Class 5" under this article's name. Handbook of Econometrics, Volume I V, Edited by R.F. Engle and D.L. McFadden © 1994 Elsevier Science B.V. All rights reserved