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  • Rasmus Søndergaard Pedersen (a1) and Anders Rahbek (a1)

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.

Corresponding author
*Address correspondence to Rasmus Søndergaard Pedersen, Department of Economics, University of Copenhagen, Øster Farimagsgade 5, 1353 Copenhagen K, Denmark; e-mail:
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Department of Economics, University of Copenhagen, Denmark. This research was supported by the Danish Council for Independent Research (DSF Grant 015-00028B). Pedersen is grateful for support from the Carlsberg Foundation. We thank the Co-Editor (Dennis Kristensen) and two referees as well as seminar participants at Cambridge, Helsinki, and Oxford Universities. We also thank participants at (EC)2 2017, Heikametrics 2017, Toulouse Financial Econometrics Conference 2018, and Brunel Conference 2018 for comments and discussions of a previous draft of the article.

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Econometric Theory
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