Groupe d’études et de recherche en analyse des décisions

Residual Processes of GARCH Models and Their Applications

Hao Yu

It is a common practice that people treat residuals of a linear time series model as if they were the original iid errors. Can the same practice be applied to residuals of nonlinear time series model such as GARCH models? Recent results have shown that the empirical processes constructed from residuals of GARCH models are no longer asymptotically parameter free. This makes some statistical inferences invalid. However we will show that it is still possible to construct residual processes as though residuals behave like iid. Applications to change-point problems and goodness-of-fit tests are considered, in particular CUSUM statistics for testing GARCH model structure change and the Jarque-Bera omnibus statistic for testing normality of the unobservable innovation distribution of a GARCH model. The use of residuals for constructing a kernel density function estimation of the innovation distribution is discussed.