Statistica Sinica 15(2005), 215-234

ASYMPTOTIC THEORY FOR ARCH-SM MODELS:

LAN AND RESIDUAL EMPIRICAL PROCESSES

Sangyeol Lee and Masanobu Taniguchi

Seoul National University and Waseda University

Abstract: In this paper, we have two asymptotic objectives: the LAN and the residual empirical process for a class of ARCH($\infty$)-SM (stochastic mean) models, which covers finite-order ARCH and GARCH models. First, we establish the LAN for the ARCH($\infty$)-SM model and, based on it, construct an asymptotically optimal test when the parameter vector contains a nuisance parameter. Also, we discuss asymptotically efficient estimators for unknown parameters when the innovation density is known and when it is unknown. For the residual empirical process, we investigate its asymptotic behavior in ARCH($q$)-SM models. We show that, unlike the usual autoregressive model, the limiting distribution in this case depends upon the estimator of the regression parameter as well as those of the ARCH parameters.

Key words and phrases: ARCH model, ARCH(∞)-SM model, asymptotically efficient estimator, asymptotically optimal test, GARCH model, Gaussian process, LAN, residual empirical process, weak convergence.