TESTING GARCH-X TYPE MODELS

We present novel theory for testing for reduction of GARCH-X type models with an exogenous (X) covariate to standard GARCH type models. To deal with the problems of potential nuisance parameters on the boundary of the parameter space as well as lack of identification under the null, we exploit a noticeable property of specific zero-entries in the inverse information of the GARCH-X type models. Specifically, we consider sequential testing based on two likelihood ratio tests and as demonstrated the structure of the inverse information implies that the proposed test neither depends on whether the nuisance parameters lie on the boundary of the parameter space, nor on lack of identification. Asymptotic theory is derived essentially under stationarity and ergodicity, coupled with a regularity assumption on the exogenous covariate X. Our general results on GARCH-X type models are applied to Gaussian based GARCH-X models, GARCH-X models with Student's t-distributed innovations as well as integer-valued GARCH-X (PAR-X) models.