{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T19:57:23Z","timestamp":1775332643505,"version":"3.50.1"},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2016,10,28]],"date-time":"2016-10-28T00:00:00Z","timestamp":1477612800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p>In this paper we derive matrix formulae in closed form for higher order moments and give sufficient conditions for higher order stationarity of Markov switching VARMA models. We provide asymptotic theory for sample higher order moments which can be used for testing multivariate normality. As an application, we propose new definitions of multivariate skewness and kurtosis measures for such models, and relate them with the existing concepts in the literature. Our work completes the statistical analysis developed in the fundamental paper of Francq and Zako\u00efan (2001,<jats:italic>Econometric Theory<\/jats:italic>18, 815\u2013818) and relates with the concepts of multivariate skewness and kurtosis proposed by Mardia (1970,<jats:italic>Biometrika<\/jats:italic>57, 519\u2013530), M\u00f3ri, Rohatgi, and Sz\u00e9kely (1993,<jats:italic>Theory of Probability and its Applications<\/jats:italic>38, 547\u2013551), and Kollo (2008,<jats:italic>Journal of Multivariate Analysis<\/jats:italic>99, 2328\u20132338). Under suitable assumptions, our results imply that the sample estimators of the skewness and kurtosis measures proposed by these authors are consistent and asymptotically normally distributed. Finally, we check our theory statements numerically via Monte Carlo simulations.<\/jats:p>","DOI":"10.1017\/s0266466616000438","type":"journal-article","created":{"date-parts":[[2016,10,28]],"date-time":"2016-10-28T07:02:04Z","timestamp":1477638124000},"page":"1502-1515","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":30,"title":["HIGHER ORDER MOMENTS OF MARKOV SWITCHING VARMA MODELS"],"prefix":"10.1017","volume":"33","author":[{"given":"Maddalena","family":"Cavicchioli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,10,28]]},"reference":[{"key":"S0266466616000438_ref19","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1137\/1138055","article-title":"On multivariate skewness and kurtosis","volume":"38","author":"M\u00f3ri","year":"1993","journal-title":"Theory of Probability and its 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