Abstract
This study considers the residual-based CUSUM test for location-scale time series models with heteroscedasticity. The estimates- and score vector-based CUSUM tests are widely used for detecting abrupt changes in time series models. However, their performance is often unsatisfactory with severe size distortions when the underlying model is complicated and the sample size is small. To circumvent this defect, the residual-based CUSUM test is suggested as an alternative. However, this test can only detect scale parameter changes and suffers severe power loss against location parameter changes. To remedy this, we introduce a modified residual-based CUSUM test and demonstrate its validity for both location and scale parameter changes. We conduct a simulation study and data analysis for illustration.
Similar content being viewed by others
References
An, H. Z., Huang, F. C. (1996). The geometrical ergodicity of nonlinear autoregressive models. Statistica Sinica, 6, 943–956.
Berkes, I., Horvath, L., Kokoszka, P. (2004). Testing for parameter constancy in GARCH(p, q) models. Statistics and Probability Letters, 70, 263–273.
Billingsley, P. (1968). Convergence of probability measure. New York: Wiley.
Brown, R. L., Durbin, J., Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time. Journal of Royal Statistical Society Series B (Methodological), 37(2), 149–192.
Chen, M., An, H. Z. (1998). A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica, 8, 505–510.
de Pooter, M., van Dijk, D. (2004). Testing for changes in volatility in heteroskedastic time series-a further examination (No. EI 2004-38).
Francq, C., Zakoïan, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 10(4), 605–637.
Franke, J., Kirch, C., Kamgaing, J. T. (2012). Changepoints in times series of counts. Journal of Time Series Analysis, 33(5), 757–770.
Gombay, E. (2008). Change detection in autoregressive time series. Journal of Multivariate Analysis, 99(3), 451–464.
Inclán, C., Tiao, G. C. (1994). Use of cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association, 89, 913–923.
Kang, J., Lee, S. (2014). Parameter change test for poisson autoregressive models. Scandinavian Journal of Statistics, 41(4), 1136–1152.
Kiefer, J. (1959). K-sample analogues of the Kolmogorov–Smirnov and Cramer-V Mises tests. The Annals of Mathematical Statistics, 30(2), 420–447.
Kim, S., Cho, S., Lee, S. (2000). On the CUSUM test for parameter changes in GARCH(1,1) models. Communications in Statistics Theory and Methods, 29(2), 445–462.
Kirch, C., Kamgaing, J. T. (2012). Testing for parameter stability in nonlinear autoregressive models. Journal of Time Series Analysis, 33(3), 365–385.
Kokoszka, P., Leipus, R. (1999). Testing for parameter changes in ARCH models. Lithuanian Mathematical Journal, 39(2), 182–195.
Kulperger, R., Yu, H. (2005). High moment partial sum processes of residuals in GARCH models and their applications. The Annals of Statistics, 33, 2395–2422.
Lee, J., Lee, S. (2015). Parameter change test for nonlinear time series models with GARCH type errors. Journal of Korean Mathematical Society, 52, 503–522.
Lee, S., Na, O. (2005). Test for parameter change in stochastic processes based on conditional least-squares estimator. Journal of Multivariate Analysis, 93, 375–393.
Lee, S., Oh, H. (2016). Parameter change test for autoregressive conditional duration models. Annals of the Institute of Statistical Mathematics, 68(3), 621–637.
Lee, S., Song, J. (2008). Test for parameter change in ARMA models with GARCH innovations. Statistics and Probability Letters, 78, 1990–1998.
Lee, S., Ha, J., Na, O., Na, S. (2003). The cusum test for parameter change in time series models. Scandinavian Journal of Statistics, 30(4), 781–796.
Lee, S., Tokutsu, Y., Maekawa, K. (2004). The cusum test for parameter change in regression models with ARCH errors. Journal of the Japan Statistical Society, 34(2), 173–188.
Meitz, M., Saikkonen, P. (2011). Parameter estimation in nonlinear AR-GARCH models. Econometric Theory, 27(6), 1236–1278.
Oh, H., Lee, S. (2017a). On score vector- and residual-based cusum tests in ARMA-GARCH models. Statistical Methods and Applications, 1, 1–22. https://doi.org/10.1007/s10260-017-0408-9.
Oh, H., Lee, S. (2017b). Bootstrap parameter change test for location-scale time series models with heteroscedasticity. (Submitted for publication).
Page, E. S. (1955). A test for change in a parameter occurring at an unknown point. Biometrika, 42(3/4), 523–527.
Straumann, D., Mikosch, T. (2006). Quasi maximum likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. The Annals of Statistics, 34(5), 2449–2495.
Acknowledgements
We would like to thank the Editor, an AE and the two referees for their careful reading and valuable comments that improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (No. 2018R1A2A2A05019433).
About this article
Cite this article
Oh, H., Lee, S. Modified residual CUSUM test for location-scale time series models with heteroscedasticity. Ann Inst Stat Math 71, 1059–1091 (2019). https://doi.org/10.1007/s10463-018-0679-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-018-0679-4