Multivariate portmanteau tests for weak multiplicative seasonal VARMA models
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Multivariate portmanteau tests for weak multiplicative seasonal VARMA models Abdoulkarim Ilmi Amir1 · Yacouba Boubacar Maïnassara1 Received: 4 April 2018 / Revised: 10 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Numerous multivariate time series encountered in real applications display seasonal behavior. In this paper we consider portmanteau tests for testing the adequacy of structural multiplicative seasonal vector autoregressive moving-average (SVARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak SVARMA). We study the asymptotic distributions of residual autocorrelations at seasonal lags of multiple of the length of the seasonal period under weak assumptions on the noise. We deduce the asymptotic distribution of the proposed multivariate portmanteau statistics, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. A set of Monte Carlo experiments and an application of U.S. monthly housing starts and housing sold are presented. Keywords Goodness-of-fit test · Quasi-maximum likelihood estimation · Portmanteau tests · Residual autocorrelation · Weak SVARMA models Mathematics Subject Classification Primary 62M10 · 62F03 · 62F05 · Secondary 91B84 · 62P05
The authors wish to acknowledge the support from the “Séries temporelles et valeurs extrêmes : théorie et applications en modélisation et estimation des risques” Projet Région (Bourgogne Franche-Comt, France) Grant No OPE-2017-0068.
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Yacouba Boubacar Maïnassara [email protected] Abdoulkarim Ilmi Amir [email protected]
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Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France
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A. Ilmi Amir, Y. Boubacar Maïnassara
1 Introduction Seasonality naturally occurs in many time series coming from various fields of study, such as meteorology, hydrology and economics, amongst others. In univariate time series, the multiplicative seasonal autoregressive moving average (SARMA), denoted SARMA( p, q)( ps , qs ), time series model is widely used in the modeling of seasonal time series [see Chap. 12 in Hipel and Ian McLeod (1994)], where s denotes the length of the seasonal period. In econometric application, the univariate SARMA framework is very restrictive. Consequently the class of multivariate models are commonly used in time series analysis and econometrics. It describes the possible cross-relationships between the time series and not only the properties of the individual time series [see Lütkepohl (2005)]. Consider a d-dimensional stationary process X = (X t )t∈Z , satisfying a multiplicative seasonal vectorial autoregressive moving average (SVARMA) representation of the form (1) Λθ0 (L s )Aθ0 (L)X t = Ψθ0 (L s )Bθ0 (L)t , ∀t ∈ Z, where the nonseasonal AR and MA operators are defined by Aθ0 (L) = A00 − p q i i while the seasonal i=1 A0i L and Bθ0 (L) = B00 − i=1
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