Estimation and Asymptotic Inference in the AR-ARCH Model

Abstract

This article studies asymptotic properties of the quasi-maximum likelihood estimator (QMLE) for the parameters in the autoregressive (AR) model with autoregressive conditional heteroskedastic (ARCH) errors. A modified QMLE (MQMLE) is also studied. This estimator is based on truncation of individual terms of the likelihood function and is related to the recent so-called self-weighted QMLE in Ling (2007b). We show that the MQMLE is asymptotically normal irrespectively of the existence of finite moments, as geometric ergodicity alone suffice. Moreover, our included simulations show that the MQMLE is remarkably well-behaved in small samples. On the other hand, the ordinary QMLE, as is well-known, requires finite fourth order moments for asymptotic normality. But based on our considerations and simulations, we conjecture that in fact only geometric ergodicity and finite second order moments are needed for the QMLE to be asymptotically normal. Finally, geometric ergodicity for AR-ARCH processes is shown to hold under mild and classic conditions on the AR and ARCH processes.

Keywords:

• ARCH
• Asymptotic theory
• Geometric ergodicity
• Modified QMLE
• QMLE

JEL Classification:

• C22
• C52

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support from Danish Social Sciences Research Council, Project Number 2114-04-001.

Notes

¹All experiments were programmed using the random-number generator of the matrix programming language Ox 3.40 of Doornik (1998) over N = 10,000 Monte Carlo replications.

²Ox code for employing both estimators discussed in this article can be obtained from the authors.