Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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.
Davidson, J. (1994). Stochastic limit theory. Oxford: Oxford University Press.
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.
Francq, C., & Zakoïan, J. M. (1998). Estimating linear representations of nonlinear processes. Journal of Statistical Planning and Inference, 68(1), 145–165.
Francq, C., & Zakoïan, J. M. (2010). GARCH models: Structure, statistical inference and financial applications. Chichester: Wiley.
Godfrey, L. G. (1979). Testing the adequacy of a time series model. Biometrika, 66, 67–72.
Hosking, J. R. M. (1980). Lagrange-multiplier tests of time-series models. Journal of the Royal Statistical Society: Series B, 42, 170–181.
Katayama, N. (2013). Proposal of robust M tests and their applications. Woking paper series F-65, Economic Society of Kansai University.
Katayama, N. (2012). Chi-squared portmanteau tests for structural VARMA models with uncorrelated errors. Journal of Time Series Analysis, 33(6), 863–872.
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.
Kiefer, N. M., Vogelsang, T. J., & Bunzel, H. (2000). Simple robust testing of regression hypotheses. Journal of Time Series Analysis, 68, 695–714.
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.
Lee, W. M. (2007). Robust \(M\) tests using kernel-based estimators with bandwidth equal to sample size. Economic Letters, 96, 295–300.
Li, W. K. (2003). Diagnostic checks in time series. Boca Raton, FL: CRC Press.
Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297–303.
Lobato, I. N. (2001). Testing that dependent process is uncorrelated. Biometrika, 96, 1066–1076.
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.
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.
Newbold, P. (1980). The equivalence of two tests of time series model adequacy. Biometrika, 67(2), 463–465.
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.
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.
Schott, J. R. (1997). Matrix analysis for statistics. New York, NY: Wiley.
Su, J. J. (2005). On the size and power of testing for no autocorrelation under weak assumptions. Biometrika, 15, 247–257.
White, H. (1994). Estimation, inference, and specification analysis. Cambridge: Cambridge University Press.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this chapter
Cite this chapter
Katayama, N. (2016). The Portmanteau Tests and the LM Test for ARMA Models with Uncorrelated Errors. In: Li, W., Stanford, D., Yu, H. (eds) Advances in Time Series Methods and Applications . Fields Institute Communications, vol 78. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6568-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4939-6568-7_6
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-6567-0
Online ISBN: 978-1-4939-6568-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)