Bernoulli

  • Bernoulli
  • Volume 19, Number 5B (2013), 2715-2749.

A test for stationarity based on empirical processes

Philip Preuß, Mathias Vetter, and Holger Dette

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Abstract

In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov–Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate $g_{T}\to0$ such that $g_{T}T^{1/2}\to\infty$, where $T$ denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.

Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2715-2749.

Dates
First available in Project Euclid: 3 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1386078618

Digital Object Identifier
doi:10.3150/12-BEJ472

Mathematical Reviews number (MathSciNet)
MR3160569

Zentralblatt MATH identifier
1281.62183

Keywords
bootstrap empirical spectral measure goodness-of-fit tests integrated periodogram locally stationary process non-stationary processes spectral density

Citation

Preuß, Philip; Vetter, Mathias; Dette, Holger. A test for stationarity based on empirical processes. Bernoulli 19 (2013), no. 5B, 2715--2749. doi:10.3150/12-BEJ472. https://projecteuclid.org/euclid.bj/1386078618


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References

  • [1] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971) 267–281. Budapest: Akadémiai Kiadó.
  • [2] Beltrão, K.I. and Bloomfield, P. (1987). Determining the bandwidth of a kernel spectrum estimate. J. Time Series Anal. 8 21–38.
  • [3] Berg, A., Paparoditis, E. and Politis, D.N. (2010). A bootstrap test for time series linearity. J. Statist. Plann. Inference 140 3841–3857.
  • [4] Brillinger, D.R. (1981). Time Series: Data Analysis and Theory. New York: McGraw-Hill.
  • [5] Chiann, C. and Morettin, P.A. (1999). Estimation of time varying linear systems. Stat. Inference Stoch. Process. 2 253–285.
  • [6] Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis. Stochastic Process. Appl. 30 69–83.
  • [7] Dahlhaus, R. (1996). On the Kullback–Leibler information divergence of locally stationary processes. Stochastic Process. Appl. 62 139–168.
  • [8] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • [9] Dahlhaus, R. (2009). Local inference for locally stationary time series based on the empirical spectral measure. J. Econometrics 151 101–112.
  • [10] Dahlhaus, R., Neumann, M.H. and von Sachs, R. (1999). Nonlinear wavelet estimation of time-varying autoregressive processes. Bernoulli 5 873–906.
  • [11] Dahlhaus, R. and Polonik, W. (2006). Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. Ann. Statist. 34 2790–2824.
  • [12] Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series. Bernoulli 15 1–39.
  • [13] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114.
  • [14] Dette, H., Preuss, P. and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. J. Amer. Statist. Assoc. 106 1113–1124.
  • [15] Dwivedi, Y. and Subba Rao, S. (2011). A test for second-order stationarity of a time series based on the discrete Fourier transform. J. Time Series Anal. 32 68–91.
  • [16] Hannan, E. and Kavalieris, L. (1986). Regression, autoregression models. J. Time Series Anal. 7 27–49.
  • [17] Kreiß, J.P. (1988). Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Fachbereich Mathematik, Univ. Hamburg.
  • [18] Kreiß, J.P. (1997). Asymptotical properties of residual bootstrap for autoregressions. Technical report, TU Braunschweig.
  • [19] Kreiss, J.P., Paparoditis, E. and Politis, D.N. (2011). On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 2103–2130.
  • [20] Neumann, M.H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 38–76.
  • [21] Palma, W. and Olea, R. (2010). An efficient estimator for locally stationary Gaussian long-memory processes. Ann. Statist. 38 2958–2997.
  • [22] Paparoditis, E. (2009). Testing temporal constancy of the spectral structure of a time series. Bernoulli 15 1190–1221.
  • [23] Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms. J. Amer. Statist. Assoc. 105 839–851.
  • [24] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. New York: Springer.
  • [25] Priestley, M.B. (1965). Evolutionary spectra and non-stationary processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 204–237.
  • [26] Sergides, M. and Paparoditis, E. (2009). Frequency domain tests of semi-parametric hypotheses for locally stationary processes. Scand. J. Stat. 36 800–821.
  • [27] Van Bellegem, S. and von Sachs, R. (2008). Locally adaptive estimation of evolutionary wavelet spectra. Ann. Statist. 36 1879–1924.
  • [28] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. New York: Springer.
  • [29] von Sachs, R. and Neumann, M.H. (2000). A wavelet-based test for stationarity. J. Time Series Anal. 21 597–613.
  • [30] Whittle, P. (1951). Hypothesis Testing in Time Series Analysis. Uppsala: Almqvist and Wiksell.
  • [31] Whittle, P. (1952). Some results in time series analysis. Skand. Aktuarietidskr. 35 48–60.