The Annals of Statistics

Nonparametric regression for locally stationary time series

Michael Vogt

Full-text: Open access

Abstract

In this paper, we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time. These models are a natural extension of time series models with time-varying coefficients. We introduce a kernel-based method to estimate the time-varying regression function and provide asymptotic theory for our estimates. Moreover, we show that the main conditions of the theory are satisfied for a large class of nonlinear autoregressive processes with a time-varying regression function. Finally, we examine structured models where the regression function splits up into time-varying additive components. As will be seen, estimation in these models does not suffer from the curse of dimensionality.

Article information

Source
Ann. Statist., Volume 40, Number 5 (2012), 2601-2633.

Dates
First available in Project Euclid: 4 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1359987532

Digital Object Identifier
doi:10.1214/12-AOS1043

Mathematical Reviews number (MathSciNet)
MR3097614

Zentralblatt MATH identifier
1373.62459

Subjects
Primary: 62G08: Nonparametric regression 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G20: Asymptotic properties

Keywords
Local stationarity nonparametric regression smooth backfitting

Citation

Vogt, Michael. Nonparametric regression for locally stationary time series. Ann. Statist. 40 (2012), no. 5, 2601--2633. doi:10.1214/12-AOS1043. https://projecteuclid.org/euclid.aos/1359987532


Export citation

References

  • [1] An, H. Z. and Huang, F. C. (1996). The geometrical ergodicity of nonlinear autoregressive models. Statist. Sinica 6 943–956.
  • [2] Bhattacharya, R. and Lee, C. (1995). On geometric ergodicity of nonlinear autoregressive models. Statist. Probab. Lett. 22 311–315.
  • [3] Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed. Lecture Notes in Statistics 110. Springer, New York.
  • [4] Chen, M. and Chen, G. (2000). Geometric ergodicity of nonlinear autoregressive models with changing conditional variances. Canad. J. Statist. 28 605–613.
  • [5] Dahlhaus, R. (1996). On the Kullback–Leibler information divergence of locally stationary processes. Stochastic Process. Appl. 62 139–168.
  • [6] Dahlhaus, R. (1996). Asymptotic statistical inference for nonstationary processes with evolutionary spectra. In Athens Conference on Applied Probability and Time Series Analysis, Vol. II (1995) (P. M. Robinson and M. Rosenblatt, eds.). Lecture Notes in Statistics 115 145–159. Springer, New York.
  • [7] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • [8] Dahlhaus, R., Neumann, M. H. and von Sachs, R. (1999). Nonlinear wavelet estimation of time-varying autoregressive processes. Bernoulli 5 873–906.
  • [9] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114.
  • [10] Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2008). Normalized least-squares estimation in time-varying ARCH models. Ann. Statist. 36 742–786.
  • [11] Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 320–346.
  • [12] Hafner, C. M. and Linton, O. (2010). Efficient estimation of a multivariate multiplicative volatility model. J. Econometrics 159 55–73.
  • [13] Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24 726–748.
  • [14] Koo, B. and Linton, O. (2012). Semiparametric estimation of locally stationary diffusion models. J. Econometrics 170 210–233.
  • [15] Kristensen, D. (2009). Uniform convergence rates of kernel estimators with heterogeneous dependent data. Econometric Theory 25 1433–1445.
  • [16] Kristensen, D. (2011). Stationary approximations of time-inhomogenous Markov chains with applications. Mimeo.
  • [17] Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490.
  • [18] Masry, E. (1996). Multivariate local polynomial regression for time series: Uniform strong consistency and rates. J. Time Series Anal. 17 571–599.
  • [19] Moulines, E., Priouret, P. and Roueff, F. (2005). On recursive estimation for time varying autoregressive processes. Ann. Statist. 33 2610–2654.
  • [20] Subba Rao, S. (2006). On some nonstationary, nonlinear random processes and their stationary approximations. Adv. in Appl. Probab. 38 1155–1172.
  • [21] Tjøstheim, D. (1990). Non-linear time series and Markov chains. Adv. in Appl. Probab. 22 587–611.
  • [22] Vogt, M. (2012). Supplement to “Nonparametric regression for locally stationary time series.” DOI:10.1214/12-AOS1043SUPP.
  • [23] Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. Ann. Statist. 38 2187–2217.
  • [24] Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696–2729.

Supplemental materials