Seminars 2008 — Abstracts

Friday, August 22

Speaker: Boris Buchmann, Monash

Title: On the Limit Experiments of Randomly Thinned GARCH(1,1) in Deficiency

Abstract: GARCH is the most prominent nonlinear time series model, both widely applied and thoroughly studied. Aggregating its only one-dimensional innovations, it is a well-known oddity that GARCH converges in law to a diffusion driven by a two-dimensional Brownian motion. As a result, the convergence is not in Le Cam's sense of deficiency and, thus, passage from discrete to continuous time is impossible in all plausible statistical decision problems. Recently an intuitively appealing continuous time version of GARCH (called COGARCH) was introduced which is driven by an only one-dimensional Levy process, thus, maintaining one of the key features of GARCH. Further investigations have shown that COGARCH occurs as a limit of GARCH models in distribution when the innovations are randomly thinned. In this talk we investigate the validity of the corresponding approximations in Le Cam's framework of deficiency. We identify the limit experiments for two kinds of sampling schemes. If the corresponding volatilities are unobservable, we show that the limit experiment is not equivalent to COGARCH in deficiency. Otherwise, if, in addition, full information observations about the volatility processes is available, then we show that the limiting experiment is generically equivalent to COGARCH.