Detection and estimation of changes in ARCH processes

Abstract

Most statistical procedures in time series analysis rely on the assumption that the observed sample has been transformed in such a way as to form a stationary sequence. It is then often assumed that such a transformed series can be well approximated by a parametric model whose parameters are to be estimated or hypotheses related to them tested. Before carrying out such inferences it is worthwhile to verify that the transformed series is indeed stationary or, if a specific parametric model is postulated, that the parameters remain constant. A classical statistical problem, which is an extension of a two sample problem to dependent data, is to test if the observations before and after a specified moment of time follow the same model. In this paper we are, however, concerned with a change-point problem in which the time of change is unknown. The task is to test if a change has occurred somewhere in the sample and, if so, to estimate the time of its occurrence. The simplest form of departure from stationarity is a change in mean at some (unknown) point in the sample. This problem has received a great deal of attention, see e.g. Csörgő and Horváth (1997). Financial returns series have, however, typically constant zero mean, but exhibit noticeable and complex changes in the spread of observation commonly referred to as clusters of volatility.