Modified Schwarz and Hannan–Quinn information criteria for weak VARMA models

  • Yacouba Boubacar MaïnassaraEmail author
  • Célestin C. Kokonendji


Numerous multivariate time series admit weak vector autoregressive moving-average (VARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be independent and identically distributed (iid). This article considers the problem of order selection of the weak VARMA models by using the information criteria. It is shown that the use of the standard information criteria are often not justified when the iid assumption on the noise is relaxed. As a consequence, we propose the modified versions of the Schwarz or Bayesian information criterion and of the Hannan and Quinn criterion for identifying the orders of weak VARMA models. Monte Carlo experiments show that the proposed modified criteria estimate the model orders more accurately than the standard ones. An illustrative application using the squared daily returns of financial series is presented.


Identification Information criterion Order selection 

Mathematics Subject Classification

Primary 62M10 91B84 Secondary 62H99 



The research of the first author is supported by a BQR (Bonus Qualité Recherche) of the Université de Franche-Comté. We sincerely thank the associated editor and the anonymous reviewers for helpful remarks. Their detailed comments led to greatly improve the presentation.


  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csáki F (eds) 2nd international symposium on information theory. Akadémia Kiado, Budapest, pp 267–281Google Scholar
  2. Alj A, Jnasson K, Mlard G (2014) The exact Gaussian likelihood estimation of time-dependent VARMA models. Comput Stat Data Anal. doi: 10.1016/j.csda.2014.07.006
  3. Bauwens L, Laurent S, Rombouts JVK (2006) Multivariate GARCH models: a survey. J Appl Econ 21:79–109MathSciNetCrossRefGoogle Scholar
  4. Boubacar Maïnassara Y (2011) Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms. J Stat Plan Inferenc 141:2961–2975MathSciNetCrossRefzbMATHGoogle Scholar
  5. Boubacar Maïnassara Y (2012) Selection of weak VARMA models by modified Akaike’s information criteria. J Time Ser Anal 33:121–130CrossRefzbMATHGoogle Scholar
  6. Boubacar Maïnassara Y, Carbon M, Francq C (2012) Computing and estimating information matrices of weak ARMA models. Comput Stat Data Anal 56:345–361MathSciNetCrossRefzbMATHGoogle Scholar
  7. Boubacar Maïnassara Y, Francq C (2011) Estimating structural VARMA models with uncorrelated but non-independent error terms. J Multivar Anal 102:496–505MathSciNetCrossRefzbMATHGoogle Scholar
  8. Brockwell PJ, Davis RA (1991) Time series: theory and methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  9. Brüggemann R, Lütkepohl H (2001) Lag selection in subset VAR models with an application to a U.S. monetary system. In: Friedmann R, Knüppel L, Lütkepohl H (eds) Econometric studies–a festschrift in honour of Joachim Frohn Münster. LIT, Berlin, pp 107–128Google Scholar
  10. Camacho M (2004) Vector smooth transition regression models for US GDP and the composite index of leading indicators. J Forecast 23:173–196CrossRefGoogle Scholar
  11. Dufour J-M, Pelletier D (2005) Practical methods for modelling weak VARMA processes: identification, estimation and specification with a macroeconomic application. Technical report, Département de sciences économiques and CIREQ, Université de Montréal, Montréal, CanadaGoogle Scholar
  12. Francq C, Zakoïan J-M (2005) Recent results for linear time series models with non independent innovations. In: Duchesne P, Rémillard B (eds) Statistical modeling and analysis for complex data problems, Chap. 12. Springer, New York, pp 241–265CrossRefGoogle Scholar
  13. Francq C, Zakoïan J-M (2010) GARCH models: structure, statistical inference and financial applications. Wiley, ChichesterCrossRefGoogle Scholar
  14. Hannan EJ, Quinn BG (1979) The determination of the order of an autoregression. J R Stat Soc B 41:190–195MathSciNetzbMATHGoogle Scholar
  15. Hannan EJ, Rissanen J (1982) Recursive estimation of mixed of autoregressive moving average order. Biometrika 69:81–94MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hurvich CM, Tsai C-L (1989) Regression and time series model selection in small samples. Biometrika 76:297–307MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hurvich CM, Tsai C-L (1993) A corrected Akaike information criterion for vector autoregressive model selection. J Time Ser Anal 14:271–279MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jeantheau T (1998) Strong consistency of estimators for multivariate ARCH models. Econom Theory 14:70–86MathSciNetCrossRefGoogle Scholar
  19. Katayama N (2012) Chi-squared portmanteau tests for structural VARMA models with uncorrelated errors. J Time Ser Anal 33:863–872Google Scholar
  20. Lütkepohl H (2005) New introduction to multiple time series analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
  21. Neath AA, Cavanaugh JE (1997) Regression and time series model selection using variants of the Schwarz information criterion. Communications in statistics-theory and methods 26:559–580MathSciNetCrossRefzbMATHGoogle Scholar
  22. R Development Core Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna.
  23. Raftery AE (1995) Bayesian model selection in social research. Sociol Methodol 25:111–163CrossRefGoogle Scholar
  24. Reinsel GC (1997) Elements of multivariate time series analysis, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  25. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464MathSciNetCrossRefzbMATHGoogle Scholar
  26. Tsay RS (1998) Testing and modeling multivariate threshold models. J Am Stat Assoc 93:1188–1202MathSciNetCrossRefzbMATHGoogle Scholar
  27. van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Yacouba Boubacar Maïnassara
    • 1
    Email author
  • Célestin C. Kokonendji
    • 1
  1. 1.Laboratoire de Mathématiques de Besançon - UMR 6623 CNRS-UFCUniversité de Franche-ComtéBesancon CedexFrance

Personalised recommendations