The Portmanteau Tests and the LM Test for ARMA Models with Uncorrelated Errors

  • Naoya KatayamaEmail author
Part of the Fields Institute Communications book series (FIC, volume 78)


In this article, we investigate the portmanteau tests and the Lagrange multiplier (LM) test for goodness of fit in autoregressive and moving average models with uncorrelated errors. Under the assumption that the error is not independent, the classical portmanteau tests and LM test are asymptotically distributed as a weighted sum of chi-squared random variables that can be far from the chi-squared distribution. To conduct the tests, we must estimate these weights using nonparametric methods. Therefore, by employing the method of Kiefer et al. (Econometrica, 68:695–714, 2000, [11]), we propose new test statistics for the portmanteau tests and the LM test. The asymptotic null distribution of these test statistics is not standard, but can be tabulated by means of simulations. In finite-sample simulations, we demonstrate that our proposed test has a good ability to control the type I error, and that the loss of power is not substantial.


Asymptotic Variance ARMA Model Recursive Estimator Lagrange Multiplier Test Asymptotic Covariance Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank the editors, and the referees, Prof. Kohtaro Hitomi, Prof. Yoshihiko Nishiyama, Prof. Eiji Kurozumi, and Prof. Katsuto Tanaka, Prof. Tim Vogelsang, Prof. Shiqing Ling, Prof. Wai Keung Li, and Prof. A. Ian McLeod for their valuable comments and suggestions. An earlier version of this paper was presented at the 23rd (EC)2-conference at Maastricht University, held on 14–15 December 2012, and Festschrift for Prof. A. I. McLeod at Western University, held on 2–3 June 2014. I would also like to thank all participants of these conferences. This work is supported by JSPS Kakenhi (Grant Nos. 26380278, 26380279), Kansai University’s overseas research program for the academic year of 2015, and Ministry of Scienece and Technology, TAIWAN for the project from MOST 104-2811-H-006-005.


  1. 1.
    Box, G. E. P., & Pierce, A. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, 65, 1509–1526.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Davidson, J. (1994). Stochastic limit theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
  3. 3.
    Francq, C., Roy, R., & Zakoïan, J. M. (2005). Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association, 100, 532–544.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Francq, C., & Zakoïan, J. M. (1998). Estimating linear representations of nonlinear processes. Journal of Statistical Planning and Inference, 68(1), 145–165.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Francq, C., & Zakoïan, J. M. (2010). GARCH models: Structure, statistical inference and financial applications. Chichester: Wiley.CrossRefzbMATHGoogle Scholar
  6. 6.
    Godfrey, L. G. (1979). Testing the adequacy of a time series model. Biometrika, 66, 67–72.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hosking, J. R. M. (1980). Lagrange-multiplier tests of time-series models. Journal of the Royal Statistical Society: Series B, 42, 170–181.Google Scholar
  8. 8.
    Katayama, N. (2013). Proposal of robust M tests and their applications. Woking paper series F-65, Economic Society of Kansai University.Google Scholar
  9. 9.
    Katayama, N. (2012). Chi-squared portmanteau tests for structural VARMA models with uncorrelated errors. Journal of Time Series Analysis, 33(6), 863–872.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kiefer, N. M., & Vogelsang, T. J. (2002). Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Journal of Time Series Analysis, 18, 1350–1366.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kiefer, N. M., Vogelsang, T. J., & Bunzel, H. (2000). Simple robust testing of regression hypotheses. Journal of Time Series Analysis, 68, 695–714.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kuan, C. M., & Lee, W. M. (2006). Robust \(M\) tests without consistent estimation of the asymptotic covariance matrix. Journal of Time Series Analysis, 101, 1264–1275.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lee, W. M. (2007). Robust \(M\) tests using kernel-based estimators with bandwidth equal to sample size. Economic Letters, 96, 295–300.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, W. K. (2003). Diagnostic checks in time series. Boca Raton, FL: CRC Press.zbMATHGoogle Scholar
  15. 15.
    Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297–303.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lobato, I. N. (2001). Testing that dependent process is uncorrelated. Biometrika, 96, 1066–1076.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lobato, I. N., Nankervis, J. C., & Savin, N. E. (2002). Testing for zero autocorrelation in the presence of statistical dependence. Biometrika, 18(3), 730–743.MathSciNetzbMATHGoogle Scholar
  18. 18.
    McLeod, A. I. (1978). On the distribution of residual autocorrelations in Box–Jenkins models. Journal of the Royal Statistical Society: Series B, 40, 296–302.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Newbold, P. (1980). The equivalence of two tests of time series model adequacy. Biometrika, 67(2), 463–465.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Phillips, P. C. B., Sun, Y., & Jin, S. (2003). Consistent HAC estimation and robust regression testing using sharp original kernels with no truncation. Cowles Foundation discussion paper no. 1407.Google Scholar
  21. 21.
    Phillips, P. C. B., Sun, Y., & Jin, S. (2007). Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. Biometrika, 137(3), 837–894.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Schott, J. R. (1997). Matrix analysis for statistics. New York, NY: Wiley.zbMATHGoogle Scholar
  23. 23.
    Su, J. J. (2005). On the size and power of testing for no autocorrelation under weak assumptions. Biometrika, 15, 247–257.Google Scholar
  24. 24.
    White, H. (1994). Estimation, inference, and specification analysis. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculty of EconomicsKansai UniversitySuitaJapan
  2. 2.Institute of International BusinessNational Cheng Kung UniversityTainan CityTaiwan, ROC

Personalised recommendations