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10/12/04 Colloquium
DR. THEOFANIS SAPATINAS
UNIVERSITY OF CYPRUS
CYPRUS
UNIVERSITY OF CYPRUS
CYPRUS
Normalised Least-Squares Estimation in Time-Varying ARCH Models
Abstract: We consider estimation in the recently proposed class of time-varying locally stationary ARCH(p) models. We define a local normalised least-squares criterion which, unlike the previously proposed local quasi-maximum likelihood criterion, has the advantage of having a tractable, explicit solution. Under minimal moment conditions, we derive the asymptotic properties of the kernel-based normalised least-squares estimator. Despite its simplicity, tractability and ease of computation, it suffers from a number of drawbacks: we identify these and propose an adaptation of the local normalised least-squares criterion which yields an improved estimator of the time-varying ARCH(p) parameters. We introduce this estimator as a two-stage scheme, which is computationally as simple to evaluate as the kernel-based normalised least-squares estimator. Under an additional mild moment condition, we show that the asymptotic properties of this two-stage kernel-based normalised least-squares estimator are similar to those of the kernel-based normalised least-squares estimator, as well as those of the kernel-based quasi-maximum likelihood estimator. Since the stationary ARCH(p) model belongs to the class of time-varying ARCH(p) models, the estimators considered here can also be used to estimate stationary ARCH(p) parameters. We summarise the normalised least-squares and two-stage normalised least-squares schemes for the estimation of stationary ARCH(p) parameters, and state their asymptotic properties. As an illustration, we use several exchange rate datasets to illustrate the forecasting ability of the time-varying ARCH(1) model whose parameters are estimated using the two-stage kernel-NLS estimator, and make comparisons with the benchmark stationary GARCH(1,1) model. Also, we fit the time-varying ARCH(1) model to the USD/GBP exchange rate series and demonstrate its goodness-of-fit by examining the residuals. (This is a joint work with Piotr Fryzlewicz (Imperial College, UK) and Suhasini Subba Rao (University of Heidelberg, Germany).)