Electronic Journal of Statistics

Discriminating between long-range dependence and non-stationarity

Philip Preuß and Mathias Vetter

Full-text: Open access

Abstract

This paper is devoted to the discrimination between a stationary long-range dependent model and a non stationary process. We develop a nonparametric test for stationarity in the framework of locally stationary long memory processes which is based on a Kolmogorov-Smirnov type distance between the time varying spectral density and its best approximation through a stationary spectral density. We show that the test statistic converges to the same limit as in the short memory case if the (possibly time varying) long memory parameter is smaller than $1/4$ and justify why the limiting distribution is different if the long memory parameter exceeds this boundary. Concerning the latter case the novel FARI($\infty$) bootstrap is introduced which provides a bootstrap-based test for stationarity which shows good empirical properties if the long memory parameter is smaller than $1/2$ which is the usual restriction in the framework of long-range dependent time series. We investigate the finite sample properties of our approach in a comprehensive simulation study and employ the new test in an analysis of two data sets.

Article information

Source
Electron. J. Statist., Volume 7 (2013), 2241-2297.

Dates
First available in Project Euclid: 19 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1379596771

Digital Object Identifier
doi:10.1214/13-EJS836

Mathematical Reviews number (MathSciNet)
MR3108814

Zentralblatt MATH identifier
1293.62201

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis 62G10: Hypothesis testing

Keywords
Bootstrap empirical spectral measure goodness-of-fit test integrated periodogram locally stationary process long memory non stationary process spectral density

Citation

Preuß, Philip; Vetter, Mathias. Discriminating between long-range dependence and non-stationarity. Electron. J. Statist. 7 (2013), 2241--2297. doi:10.1214/13-EJS836. https://projecteuclid.org/euclid.ejs/1379596771


Export citation

References

  • Adak, S. (1998). Time-dependent spectral analysis of nonstationary time series., Journal of the American Statistical Association, 93:1488–1501.
  • Akaike, H. (1973)., Information Theory and an Extension of the Maximum Likelihood Principle. Budapest, Akademia Kiado, 267–281.
  • Beran, J. (2009). On parameter estimation for locally stationary long-memory processes., Journal of Statistical Planning and Inference, 139:900–915.
  • Berg, A., Paparoditis, E., and Politis, D. N. (2010). A bootstrap test for time series linearity., Journal of Statistical Planning and Inference, 140:3841–3857.
  • Berkes, I., Horvarth, L., Kokoszka, P., and Shao, Q. M. (2006). On discriminating between long-range dependence and changes in mean., Annals of Statistics, 34:1140–1165.
  • Brillinger, D. R. (1981)., Time Series: Data Analysis and Theory. McGraw Hill, New York.
  • Brockwell, P. J. and Davis, R. A. (1991)., Time Series: Theory and Methods. Springer Verlag, New York.
  • Can, S. U., Mikosch, T., and Samorodnitksy, G. (2010). Weak convergence of the function-indexed integrated periodogram for infinite variance processes., Bernoulli, 17(4):995–1015.
  • Chang, C. and Morettin, P. (1999). Estimation of time-varying linear systems., Statistical Inference for Stochastic Processes, 2:253–285.
  • Chen, Y., Härdle, W., and Pigorsch, U. (2010). Localized realized volatility modeling., Journal of the American Statistical Association, 105(492):1376–1393.
  • Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis., Stochastic Process and Their Applications, 30:69–83.
  • Dahlhaus, R. (1997). Fitting time series models to nonstationary processes., Annals of Statistics, 25(1):1–37.
  • Dahlhaus, R. (2009). Local inference for locally stationary time series based on the empirical spectral measure., Journal of Econometrics, 151:101–112.
  • Dahlhaus, R. and Polonik, W. (2006). Nonparametric quasi maximum likelihood estimation for Gaussian locally stationary processes., Annals of Statistics, 34(6):2790–2824.
  • Dahlhaus, R. and Polonik, W. (2009). Empirical spectral processes for locally stationary time series., Bernoulli, 15:1–39.
  • Dehling, H., Rooch, A., and Taqqu, M. S. (2013). Nonparametric change-point tests for long-range dependent data., Scandinavian Journal of Statistics, 40:153–173.
  • Dette, H., Preuß, P., and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing., Journal of the American Statistical Association, 106(495):1113–1124.
  • Doukhan, P., Oppenheim, G., and Taqqu, M. S. (2002)., Theory and Applications of Long-Range Dependence. Birkhäuser, Boston.
  • Dwivedi, Y. and Subba Rao, S. (2010). A test for second order stationarity of a time series based on the discrete fourier transform., Journal of Time Series Analysis, 32(1):68–91.
  • Eichler, M. (2008). Testing nonparametric and semiparametric hypotheses in vector stationary processes., Journal of Multivariate Analysis, 99:968–1009.
  • Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series., Ann. Statist., 14(2):517–532.
  • Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence., Probability Theory and Related Fields, 74:213–240.
  • Fryzlewicz, P., Sapatinas, T., and Subba Rao, S. (2006). A Haar-Fisz technique for locally stationary volatility estimation., Biometrika, 93:687–704.
  • Giraitis, L., Koul, H. L., and Surgailis, D. (2012)., Large Sample Inference for Long Memory Processes. Imperial College Press.
  • Granger, C. W. J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing., Journal of Time Series Analysis, 1:15–29.
  • Hannan, E. J. and Kavalieris, L. (1986). Regression, autoregression models., J. Time Ser. Anal., 7(1):27–49.
  • Henry, M. and Zaffaroni, P. (2002). The long-range dependence paradigm for macroeconomics and finance. In, Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. S. Taqqu). Birkhäuser, Boston, 417–438
  • Hosking, J. R. M. (1981). Fractional differencing., Biometrika, 68:165–176.
  • Kokoszka, P. and Mikosch, T. (1997). The integrated periodogram for long-memory processes with finite or infinite variance., Stochastic Process. Appl., 66(1):55–78.
  • Kokoszka, P. S. and Taqqu, M. S. (1995). Fractional ARIMA with stable innovations., Stochastic Processes and Their Applications, 60:19–47.
  • Kreiß, J.-P. (1988)., Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Fachbereich Mathematik, Universität Hamburg.
  • Kreiß, J.-P. and Paparoditis, E. (2011). Bootstrapping locally stationary processes. Technical, report.
  • Kreiß, J.-P., Paparoditis, E., and Politis, D. N. (2011). On the range of the validity of the autoregressive sieve bootstrap., Annals of statistics, 39(4):2103–2130.
  • Lavancier, F., Leipus, R., Philippe, A., and Surgailis, D. (2011). Detection of non-constant long memory parameter. Preprint, Université de, Nantes.
  • Lifshits, M. (1984). Absolute continuity of functionals of “supremum” type for Gaussian processes., Journal of Soviet Mathematics, 27:3103–3112.
  • Mikosch, T. and Starica, C. (2004). Non-stationarities in financial time series, the long range dependence and the IGARCH effects., The Review of Economics and Statistics, 86:378–390.
  • Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and applications to adaptive estimation of evolutionary spectra., Annals of Statistics, 25:38–76.
  • Newey, W. K. (1991). Uniform convergence in probability and stochastic equicontinuity., Econometrica, 59(4):1161–1167.
  • Palma, W. (2007)., Long-Memory Time Series: Theory and Methods. Wiley Series in Probability and Statistics.
  • Palma, W. and Olea, R. (2010). An efficient estimator for locally stationary Gaussian long-memory processes., Annals of Statistics, 38(5):2958–2997.
  • Paparoditis, E. (2009). Testing temporal constancy of the spectral structure of a time series., Bernoulli, 15:1190–1221.
  • Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms., Journal of the American Statistical Association, 105(490):839–851.
  • Park, K. and Willinger, W. (2000)., Self-Similar Network Traffic and Performance Evaluation. Wiley, New York.
  • Perron, P. and Qu, Z. (2010). Long-memory and level shifts in the volatility of stock market return indices., Journal of Business and Economic Statistics, 23(2):275–290.
  • Preuß, P., Vetter, M., and Dette, H. (2012). A test for stationarity based on empirical processes., To appear in Bernoulli.
  • Roueff, F. and von Sachs, R. (2011). Locally stationary long memory estimation., Stochastic Processes and Their Applications, 121:813–844.
  • Sakiyama, K. and Taniguchi, M. (2004). Discriminant analysis for locally stationary processes., Journal of Multivariate Analysis, 90:282–300.
  • Sibbertsen, P. and Kruse, R. (2009). Testing for a change in persistence under long-range dependencies., Journal of Time Series Analysis, 30:263–285.
  • Starica, C. and Granger, C. (2005). Nonstationarities in stock returns., The Review of Economics and Statistics, 87:503–522.
  • Van Bellegem, S. and von Sachs, R. (2008). Locally adaptive estimation of evolutionary wavelet spectra., Annals of Statistics, 36(4):1879–1924.
  • van der Vaart, A. and Wellner, J. (1996)., Weak Convergence and Empirical Processes. Springer, Berlin.
  • von Sachs, R. and Neumann, M. H. (2000). A wavelet-based test for stationarity., Journal of Time Series Analysis, 21:597–613.
  • Whittle, P. (1951)., Hypothesis Testing in Time Series Analysis. HUppsala: Almqvist and Wiksell.
  • Zygmund, A. (1959)., Trigonometric Series I. Cambridge University Press.