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HIGHER ORDER MOMENTS OF MARKOV SWITCHING VARMA MODELS

Published online by Cambridge University Press:  28 October 2016

Maddalena Cavicchioli
Maddalena Cavicchioli*
Affiliation:
University of Modena and Reggio Emilia
*
*Address correspondence to Maddalena Cavicchioli, Department of Economics “Marco Biagi,” University of Modena and Reggio Emilia, Viale Berengario 51, 41121 Modena, Italy; e-mail: maddalena.cavicchioli@unimore.it.
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Abstract

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 Zakoïan (2001, Econometric Theory 18, 815–818) and relates with the concepts of multivariate skewness and kurtosis proposed by Mardia (1970, Biometrika 57, 519–530), Móri, Rohatgi, and Székely (1993, Theory of Probability and its Applications 38, 547–551), and Kollo (2008, Journal of Multivariate Analysis 99, 2328–2338). 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.

Type
MISCELLANEA
Information
Econometric Theory , Volume 33 , Issue 6 , December 2017 , pp. 1502 - 1515
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

I am very grateful to Professor Zakoïan (CREST, France) who suggested the interesting problem of computation of higher order moments for MS VARMA models during my PhD defence, held in March 2014 at the University of Venice, Italy. I would like to thank the Editor of the journal, Professor Peter C.B. Phillips, the Co-Editor, Professor Giuseppe Cavaliere, and the three anonymous referees for their constructive comments and useful suggestions which were most valuable for improvement of the final version of the paper.

References

REFERENCES

Cavicchioli, M. (2013) Spectral density of Markov-switching VARMA models. Economics Letters 121, 218–220.CrossRefGoogle Scholar
Cavicchioli, M. (2014) Determining the number of regimes in Markov switching VAR and VMA models. Journal of Time Series Analysis 35, 173–186.CrossRefGoogle Scholar
Costa, O.L.V., Fragoso, M.D., & Marques, R.P. (2005) Discrete-Time Markov Jump Linear Systems. Probability and Its Applications. Springer-Verlag.CrossRefGoogle Scholar
Enomoto, R., Okamoto, N., & Seo, T. (2012) Multivariate normality test using Srivastava’s skewness and kurtosis. SUT Journal of Mathematics 48, 103–115.Google Scholar
Francq, C. & Zakoïan, J.M. (2001) Stationarity of multivariate Markov switching ARMA models. Journal of Econometrics 102, 339–364.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.M. (2002) Comments on the paper by Minxian Yang: Some properties of vector autoregressive processes with Markov switching coefficients. Econometric Theory 18, 815–818.CrossRefGoogle Scholar
Hamilton, J.D. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.CrossRefGoogle Scholar
Hamilton, J.D. (1990) Analysis of time series subject to changes in regime. Journal of Econometrics 45, 39–70.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.Google Scholar
Henze, N. (2002) Invariant tests for multivariate normality: A critical review. Statistical Papers 43, 467–506.CrossRefGoogle Scholar
Horn, R.A. & Johnson, C.R. (1985) Matrix Analysis. Cambridge University Press.Google Scholar
Jones, G.L. (2004) On the Markov chain central limit theorem. Probability Surveys 1, 299–320.CrossRefGoogle Scholar
Kollo, T. (2008) Multivariate skewness and kurtosis measures with an application in ICA. Journal of Multivariate Analysis 99, 2328–2338.CrossRefGoogle Scholar
Krishnamurthy, V. & Ryden, T. (1998) Consistent estimation of linear and non-linear autoregressive models with Markov regime. Journal of Time Series Analysis 9(3), 291–307.CrossRefGoogle Scholar
Krolzig, H.M. (1997) Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag.CrossRefGoogle Scholar
Magnus, J.R. & Neudecker, H. (1979) The commutation matrix: Some properties and applications. The Annals of Statistics 7, 381–394.CrossRefGoogle Scholar
Magnus, J.R. & Neudecker, H. (1986) Symmetry, 0 – 1 matrices and Jacobians: A review. Econometric Theory 2, 157–190.Google Scholar
Mardia, K.V. (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530.CrossRefGoogle Scholar
Móri, T.F., Rohatgi, V.K., & Székely, G.J. (1993) On multivariate skewness and kurtosis. Theory of Probability and its Applications 38, 547–551.Google Scholar
Neudecker, H. & Wansbeek, T. (1987) Fourth-order properties of normally distributed random matrices. Linear Algebra and its Applications 97, 13–21.CrossRefGoogle Scholar
Pataracchia, B. (2011) The spectral representation of Markov switching ARMA models. Economics Letters 112, 11–15.CrossRefGoogle Scholar
Stelzer, R. (2009) On Markov-switching ARMA processes–stationarity, existence of moments and geometric ergodicity. Econometric Theory 25(1), 43–62.CrossRefGoogle Scholar
Thode, H.C. (2002) Testing for Normality. Statistics Textbooks and Monographs, vol. 164. Marcel Dekker.CrossRefGoogle Scholar
Timmermann, A. (2000) Moments of Markov switching models. Journal of Econometrics 96, 75–111.CrossRefGoogle Scholar
Yang, M. (2000) Some properties of vector autoregressive processes with Markov-switching coefficients. Econometric Theory 16, 23–43.CrossRefGoogle Scholar
Zhang, J. & Stine, R.A. (2001) Autocovariance structure of Markov regime switching models and model selection. Journal of Time Series Analysis 22(1), 107–124.CrossRefGoogle Scholar
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