Annals of the Institute of Statistical Mathematics

, Volume 57, Issue 3, pp 575–595 | Cite as

Testing for serial correlation of unknown form in cointegrated time series models

  • Pierre Duchesne
Time Series


Portmanteau test statistics are useful for checking the adequacy of many time series models. Here we generalized the omnibus procedure proposed by Duchesne and Roy (2004,Journal of Multivariate Analysis,89, 148–180) for multivariate stationary autoregressive models with exogenous variables (VARX) to the case of cointegrated (or partially nonstationary) VARX models. We show that for cointegrated VARX time series, the test statistic obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator, is asymptotically standard normal. The parameters of the model can be estimated by conditional maximum likelihood or by asymptotically equivalent estimation procedures. The procedure relies on a truncation point or a smoothing parameter. We state conditions under which the asymptotic distribution of the test statistic is unaffected by a data-dependent method. The finite sample properties of the test statistics are studied via a small simulation study.

Key words and phrases

Vector autoregressive process cointegration exogenous variables kernel spectrum estimator diagnostic test portmanteau test 


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  1. Ahn, S. K. and Reinsel, G. C. (1988). Nested reduced-rank autoregressive models for multiple time series,Journal of the American Statistical Association,83, 849–856.MATHMathSciNetCrossRefGoogle Scholar
  2. Ahn, S. K. and Reinsel, G. C. (1990). Estimation for partially nonstationary multivariate autoregressive models,Journal of the American Statistical Association,85, 813–823.MATHMathSciNetCrossRefGoogle Scholar
  3. Beltrao, K. and Bloomfield, P. (1987). Determining the bandwidth of a kernel spectrum estimate,Journal of Time Series Analysis,8, 21–38.MATHMathSciNetGoogle Scholar
  4. Duchesne, P. and Roy, R. (2004). On consistent testing for serial correlation of unknown form in vector time series models,Journal of Multivariate Analysis,89, 148–180.MATHMathSciNetCrossRefGoogle Scholar
  5. Engle, R. F. and Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing,Econometrica,55, 251–276.MATHMathSciNetCrossRefGoogle Scholar
  6. Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification,Journal of Econometrics,16, 121–130.CrossRefGoogle Scholar
  7. Granger, C. W. J. and Weiss, A. A. (1983). Time series analysis of error-correction models,Studies in Econometrics, Time Series, and Multivariate Statistics, 255–278, Academic Press, New York.Google Scholar
  8. Hannan, E. J. (1970).Multiple Time Series, Wiley, New York.MATHGoogle Scholar
  9. Hannan, E. J. and Deistler, M. (1988).The Statistical Theory of Linear Systems, Wiley, New York.MATHGoogle Scholar
  10. Hong, Y. and Shehadeh, R. D. (1999). A new test for ARCH effects and its finite-sample performance,Journal of Business and Economic Statistics,17 91–108.MathSciNetCrossRefGoogle Scholar
  11. Hosking, J. (1980). The multivariate portmanteau statistic,Journal of the American Statistical Association,75, 602–608.MATHMathSciNetCrossRefGoogle Scholar
  12. Judge, G. G., Hill, R. C., Griffiths, W. E., Lütkepohl, H. and Lee, T.-C. (1985).The Theory and Practice of Econometrics, 2nd ed., Wiley, New York.MATHGoogle Scholar
  13. Lütkepohl, H. (1982). Differencing multiple time series: Another look at Canadian money and income data,Journal of Time Series Analysis,3, 235–243.MATHMathSciNetGoogle Scholar
  14. Lütkepohl, H. (1993).Introduction to Multiple Time Series Analysis, 2nd ed., Springer-Verlag, Berlin.Google Scholar
  15. Pham, D. T., Roy, R. and Cédras, L. (2003). Tests for non-correlation of two cointegrated ARMA time series,Journal of Time Series Analysis,24, 553–577.MATHMathSciNetCrossRefGoogle Scholar
  16. Phillips, P. C. B. and Durlauf, S. N. (1986). Multiple time series regression with integrated processes,Review of Economic Studies,53, 473–495.MATHMathSciNetCrossRefGoogle Scholar
  17. Priestley, M. B. (1981).Spectral Analysis and Time Series, Vol. 1: Univariate Series, Academic Press, New York.Google Scholar
  18. Reinsel, G. C. and Ahn, S. K. (1992). Vector AR models with unit roots and reduced rank structure: Estimation, likelihood ratio test, and forecasting,Journal of Time Series Analysis,13, 353–375.MATHMathSciNetGoogle Scholar
  19. Robinson, P. M. (1991). Automatic frequency domain inference on semiparametric and non-parametric models,Econometrica,59, 1329–1363.MATHMathSciNetCrossRefGoogle Scholar
  20. Robinson, P. M. (1994). Time series with strong dependence,Advances in Econometrics, Sixth World Congress, Vol. 1 (ed. C. Sims), Cambridge University Press, 47–95.Google Scholar
  21. Sims, C. A., Stock, J. H. and Watson, M. W. (1990). Inference in linear time series models with some unit roots.Econometrica,58, 113–144.MATHMathSciNetCrossRefGoogle Scholar
  22. Yap, S. F. and Reinsel, G. C. (1995). Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model,Journal of the American Statistical Association,90, 253–267.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics 2005

Authors and Affiliations

  • Pierre Duchesne
    • 1
  1. 1.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

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