## The Annals of Statistics

### Gradient-based structural change detection for nonstationary time series M-estimation

#### Abstract

We consider structural change testing for a wide class of time series M-estimation with nonstationary predictors and errors. Flexible predictor-error relationships, including exogenous, state-heteroscedastic and autoregressive regressions and their mixtures, are allowed. New uniform Bahadur representations are established with nearly optimal approximation rates. A CUSUM-type test statistic based on the gradient vectors of the regression is considered. In this paper, a simple bootstrap method is proposed and is proved to be consistent for M-estimation structural change detection under both abrupt and smooth nonstationarity and temporal dependence. Our bootstrap procedure is shown to have certain asymptotically optimal properties in terms of accuracy and power. A public health time series dataset is used to illustrate our methodology, and asymmetry of structural changes in high and low quantiles is found.

#### Article information

Source
Ann. Statist., Volume 46, Number 3 (2018), 1197-1224.

Dates
Revised: February 2017
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.aos/1525313080

Digital Object Identifier
doi:10.1214/17-AOS1582

Mathematical Reviews number (MathSciNet)
MR3798001

Zentralblatt MATH identifier
1392.62280

#### Citation

Wu, Weichi; Zhou, Zhou. Gradient-based structural change detection for nonstationary time series M-estimation. Ann. Statist. 46 (2018), no. 3, 1197--1224. doi:10.1214/17-AOS1582. https://projecteuclid.org/euclid.aos/1525313080

#### References

• [1] Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61 821–856.
• [2] Arcones, M. A. (1996). The Bahadur–Kiefer representation of $L_{p}$ regression estimators. Econometric Theory 12 257–283.
• [3] Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1–16.
• [4] Babu, G. J. (1989). Strong representations for LAD estimators in linear models. Probab. Theory Related Fields 83 547–558.
• [5] Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Stat. 37 577–580.
• [6] Bai, J. (1996). Testing for parameter constancy in linear regressions: An empirical distribution function approach. Econometrica 64 597–622.
• [7] Bai, J. and Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66 47–78.
• [8] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327.
• [9] Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time. J. Roy. Statist. Soc. Ser. B 37 149–192.
• [10] Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc. 95 888–902.
• [11] Dette, H., Wu, W. and Zhou, Z. (2015). Change point analysis of second order characteristics in non-stationary time series. Available at arXiv:1503.08610.
• [12] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007.
• [13] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491–1518.
• [14] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand. J. Stat. 27 715–731.
• [15] Hansen, B. E. (2000). Testing for structural change in conditional models. J. Econometrics 97 93–115.
• [16] He, X. and Zhu, L.-X. (2003). A lack-of-fit test for quantile regression. J. Amer. Statist. Assoc. 98 1013–1022.
• [17] Juhl, T. and Xiao, Z. (2009). Tests for changing mean with monotonic power. J. Econometrics 148 14–24.
• [18] Kiefer, J. (1967). On Bahadur’s representation of sample quantiles. Ann. Math. Stat. 38 1323–1342.
• [19] Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer, New York.
• [20] Mccabe, B. P. M. and Harrison, M. J. (1980). Testing the constancy of regression relation-ships over time using least squares residuals. J. R. Stat. Soc. Ser. C. Appl. Stat. 29 142–148.
• [21] Ploberger, W. and Krämer, W. (1992). The CUSUM test with OLS residuals. Econometrica 60 271–285.
• [22] Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York.
• [23] Portnoy, S. (1991). Asymptotic behavior of regression quantiles in nonstationary, dependent cases. J. Multivariate Anal. 38 100–113.
• [24] Powell, J. L. (1991). Estimation of monotonic regression models under quantile restrictions. In Nonparametric and Semiparametric Methods in Econometrics and Statistics (Durham, NC, 1988). Internat. Sympos. Econom. Theory Econometrics 357–384. Cambridge Univ. Press, Cambridge.
• [25] Prášková, Z. and Chochola, O. (2014). M-procedures for detection of a change under weak dependence. J. Statist. Plann. Inference 149 60–76.
• [26] Qu, Z. (2008). Testing for structural change in regression quantiles. J. Econometrics 146 170–184.
• [27] Su, L. and White, H. (2010). Testing structural change in partially linear models. Econometric Theory 26 1761–1806.
• [28] Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. Oxford Univ. Press, New York.
• [29] Wu, W. and Zhou, Z. (2018). Supplement to “Gradient-based structural change detection for nonstationary time series M-estimation.” DOI:10.1214/17-AOS1582SUPP.
• [30] Wu, W. B. (2007). $M$-estimation of linear models with dependent errors. Ann. Statist. 35 495–521.
• [31] Zhang, T. and Wu, W. B. (2012). Inference of time-varying regression models. Ann. Statist. 40 1376–1402.
• [32] Zhou, Z. (2013). Heteroscedasticity and autocorrelation robust structural change detection. J. Amer. Statist. Assoc. 108 726–740.
• [33] Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 513–531.

#### Supplemental materials

• Supplement to “Gradient-based structural change detection for nonstationary time series M-estimation”. We provide (a) detailed proofs of the theorems and lemmas, (b) theoretical investigation on parameter estimation under the alternative hypothesis, (c) the analysis of dynamic models, (d) extension of our methodology to finitely many M-estimations and (e) extra simulation results.