The Portmanteau Tests and the LM Test for ARMA Models with Uncorrelated Errors
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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, ), 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.
KeywordsAsymptotic Variance ARMA Model Recursive Estimator Lagrange Multiplier Test Asymptotic Covariance Matrix
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.
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