HIGHER ORDER ASYMPTOTIC THEORY WHEN A PARAMETER IS ON A BOUNDARY WITH AN APPLICATION TO GARCH MODELS

Abstract

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.