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Comparison Results for GARCH Processes

Part of: Distribution theory - Probability Mathematical finance

Published online by Cambridge University Press:  30 January 2018

Fabio Bellini ,
Franco Pellerey ,
Carlo Sgarra  and
Salimeh Yasaei Sekeh
Fabio Bellini*
Affiliation:
Universitá di Milano Bicocca
Franco Pellerey*
Affiliation:
Politecnico di Torino
Carlo Sgarra*
Affiliation:
Politecnico di Milano
Salimeh Yasaei Sekeh*
Affiliation:
University of Bojnord
*
∗ Postal address: Dipartimento di Metodi Quantitativi, Universitá di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. Email address: fabio.bellini@unimib.it
∗∗ Postal address: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy. Email address: franco.pellerey@polito.it
∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy. Email address: carlo.sgarra@polimi.it
∗∗∗∗ Postal address: Department of Mathematics, University of Bojnord, Bojnord, Iran. Email address: sa_yasaei@yahoo.com
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Abstract

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We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

Keywords

GARCHconvex orderpeakednesskurtosissupermodularity

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 91G70: Statistical methods, econometrics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 685 - 698
Copyright
© Applied Probability Trust 

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