A Bootstrap View on Dickey-Fuller Control Charts for AR(1) Series


  • Ansgar Steland RWTH Aachen University, Germany




Dickey-Fuller control charts aim at monitoring a time series until a given time horizon to detect stationarity as early as possible. That problem appears in many fields, especially in econometrics and the analysis of economic equilibria. To improve upon asymptotic control limits (critical values), we study the bootstrap and establish its a.s. consistency for fixed alternatives. Simulations indicate that the bootstrap control chart works very well.


Billingsley, P. (1968). Convergence of probability measures. New York: Wiley and Sons.

Chan, N. H., and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics, 16, 367-401.

Dickey, D. A., and Fuller, W. A. (1979). Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-431.

Ferger, D. (1995). Nonparametric tests for nonstandard change-point problems. Annals of Statistics, 23, 1848-1861.

Fuller, W. A. (1976). Introduction to Statistical Time Series. Wiley.

Granger, C.W. J. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 121-130.

Hušková, M. (1999). Gradual change versus abrupt change. Journal of Statistical Planning and Inference, 76, 109-125.

Hušková, M., and Slabý, A. (2001). Permutation tests for multiple changes. Kybernetika, 37, 605-622.

Jacod, J., and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes. Springer.

Park, J. Y. (2003). Bootstrap unit root tests. Econometrica, 71, 1845-1895.

Pawlak, M., Rafajlowicz, E., and Steland, A. (2004). Detecting jumps in time series - Nonparametric setting. Journal of Nonparametric Statistics, 16, 329-347.

Rao, M. M. (1978). Asymptotic distribution of an estimator of the boundary parameter of an unstable process. Annals of Statistics, 6, 185-190.

Steland, A. (2004). Sequential control of time series by functionals of kernel-weighted empirical processes under local alternatives. Metrika, 60, 229-249.

Steland, A. (2005a). On detection of unit roots generalising the classic Dickey-Fuller approach. In Proceedings of the 5th st. petersburg workshop on simulations (p. 653-658).

Steland, A. (2005b). Optimal sequential kernel smoothers under local nonparametric alternatives for dependent processes. Journal of Statistical Planning and Inference, 132, 131-147.

Steland, A. (2005c). Random walks with drift - A sequential view. Journal of Time Series Analysis, 26, 917-942.

White, J. S. (1958). The limiting distribution of the serial coefficient in the explosive case. Annals of Mathematical Statistics, 29, 1188-1197.




How to Cite

Steland, A. (2016). A Bootstrap View on Dickey-Fuller Control Charts for AR(1) Series. Austrian Journal of Statistics, 35(2&3), 339–346. https://doi.org/10.17713/ajs.v35i2&3.381