Abstract

For high-dimensional time series, we propose a new test to detect

white noise that is not necessarily assumed to be independent and identically

distributed.

The test can be viewed as a modified portmanteau test in high

dimensions, and the critical value of the test statistic is approximated by a multiplier bootstrap method. We provide asymptotic properties of our test under

the null hypothesis. The usefulness of our tests is demonstrated by simulations

and one real example, particularly for detecting dense alternatives.

Information

Preprint No.SS-2024-0037
Manuscript IDSS-2024-0037
Complete AuthorsZeren Zhou, Min Chen
Corresponding AuthorsMin Chen
Emailsmchen@amss.ac.cn

References

  1. Bai, J. and S. Ng (2002). Determining the number of factors in approximate factor models. Econometrica 70(1), 191–221.
  2. Baltagi, B. H., C. Kao, and F. Wang (2021). Estimating and testing high dimensional factor models with multiple structural changes. Journal of Econometrics 220(2), 349–365.
  3. Box, G. E. and D. A. Pierce (1970). Distribution of residual autocorrelations in autoregressiveintegrated moving average time series models. Journal of the American statistical Association 65(332), 1509–1526.
  4. Chang, J., Q. Yao, and W. Zhou (2017). Testing for high-dimensional white noise using maximum cross-correlations. Biometrika 104(1), 111–127.
  5. Chitturi, R. V. (1974). Distribution of residual autocorrelations in multiple autoregressive schemes. Journal of the American Statistical Association 69(348), 928–934.
  6. Daniel Peña, E. S. and V. J. Yohai (2019). Forecasting multiple time series with one-sided dynamic principal components. Journal of the American Statistical Association 114(528), 1683–1694.
  7. Fan, J., Y. Liao, and J. Yao (2015). Power enhancement in high-dimensional cross-sectional tests. Econometrica 83(4), 1497–1541.
  8. Fan, J., K. Wang, Y. Zhong, and Z. Zhu (2021). Robust High-Dimensional Factor Models with Applications to Statistical Machine Learning. Statistical Science 36(2), 303–327.
  9. Gallagher, C. M. and T. J. Fisher (2015). On weighted portmanteau tests for time-series goodness-of-fit. Journal of Time Series Analysis 36(1), 67–83.
  10. He, Y., G. Xu, C. Wu, and W. Pan (2021). Asymptotically independent u-statistics in highdimensional testing. The Annals of Statistics 49(1), 154–181.
  11. Hong, Y. (1996). Consistent testing for serial correlation of unknown form. Econometrica 64(4), 837–864.
  12. Hosking, J. R. (1980). The multivariate portmanteau statistic. Journal of the American Statistical Association 75(371), 602–608.
  13. Lam, C. and Q. Yao (2012). Factor modeling for high-dimensional time series: Inference for the number of factors. The Annals of Statistics 40(2), 694–726.
  14. Li, M. and Y. Zhang (2022). Bootstrapping multivariate portmanteau tests for vector autoregressive models with weak assumptions on errors. Computational Statistics & Data Analysis 165, 107321.
  15. Li, W. and A. McLeod (1981). Distribution of the residual autocorrelations in multivariate arma time series models. Journal of the Royal Statistical Society: Series B (Methodological) 43(2), 231–239.
  16. Li, Z., C. Lam, J. Yao, and Q. Yao (2019). On testing for high-dimensional white noise. The Annals of Statistics 47(6), 3382–3412.
  17. Ling, S., R. S. Tsay, and Y. Yang (2021). Testing serial correlation and arch effect of highdimensional time-series data. Journal of Business & Economic Statistics 39(1), 136–147.
  18. Ljung, G. M. and G. E. Box (1978, 08). On a measure of lack of fit in time series models. Biometrika 65(2), 297–303.
  19. Mukherjee, K. (2020). Bootstrapping m-estimators in generalized autoregressive conditional heteroscedastic models. Biometrika 107(3), 753–760.
  20. Tsay, R. S. (2020). Testing serial correlations in high-dimensional time series via extreme value theory. Journal of Econometrics 216(1), 106–117.
  21. Wang, L., E. Kong, and Y. Xia (2022). Bootstrap tests for high-dimensional white-noise. Journal of Business & Economic Statistics 41(1), 241–254.
  22. Wang, L., B. Peng, and R. Li (2015). A high-dimensional nonparametric multivariate test for mean vector. Journal of the American Statistical Association 110(512), 1658–1669.
  23. Wang, R. and X. Shao (2020). Hypothesis testing for high-dimensional time series via selfnormalization. The Annals of Statistics 48(5), 2728–2758.
  24. Xu, M., D. Zhang, and W. B. Wu (2019). Pearson’s chi-squared statistics: approximation theory and beyond. Biometrika 106(3), 716–723.
  25. Zhang, X. and G. Cheng (2018). Gaussian approximation for high dimensional vector under physical dependence. Bernoulli 24(4A), 2640–2675.
  26. Zhu, K. (2016). Bootstrapping the portmanteau tests in weak auto-regressive moving average models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 78(2), 463–485.
  27. Zhu, K. (2019). Statistical inference for autoregressive models under heteroscedasticity of unknown form. The Annals of Statistics 47(6), 3185–3215.
  28. Zhu, K. and W. K. Li (2015). A bootstrapped spectral test for adequacy in weak arma models. Journal of Econometrics 187(1), 113–130.
  29. Zhu, Q., R. Zeng, and G. Li (2020). Bootstrap inference for garch models by the least absolute deviation estimation. Journal of Time Series Analysis 41(1), 21–40. 1School of Statistics, Capital University of Economics and Business, Beijing 100070, P.R.China

Acknowledgments

The authors would like to thank the editor, the associate editor, and two

anonymous reviewers for their constructive comments and suggestions that

led to significant improvements in the paper.

We acknowledge the Key

Laboratory of Complex Systems and Data Science of the Ministry of Education for partial support. This research is partially supported by the Youth

Academic Innovation Team Construction project of Capital University of

Economics and Business (Grant No.QNTD202303).

Supplementary Materials

Additional details and supporting information can be found in the supplementary file online. In Section S1, we provide additional simulation results

of the white noise test for fitted residuals. Section S2 contains proofs for

the theoretical results outlined in the main text.

Supplementary materials are available for download.