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

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+