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Fitting time series models to nonstationary processes. (English) Zbl 0871.62080

Stationarity has always played a major role in the theoretical treatment of time series procedures. For example, the spectral density is defined for stationary processes and the important ARMA model is a stationary time series model. Furthermore, the assumption of stationarity is the basis for a general asymptotic theory: it guarantees that the increase of the sample size leads to more and more information of the same kind which is basic for an asymptotic theory to make sense. On the other hand, many series show a nonstationary behavior (e.g., in economics or sound analysis). Special techniques (such as taking differences or the consideration of the data on small time intervals) have been applied to make an analysis with stationary techniques possible.
If one abandons the assumption of stationarity, the number of possible models for time series data explodes. For example, one may consider ARMA models with time varying coefficients. In that case the time behavior of the coefficients may again be modeled in different ways. Therefore, we try to consider in this paper a general class of nonstationary processes together with a general estimation method which is a generalization of P. Whittle’s [Ark. Mat. 2, 423-434 (1953; Zbl 0053.41003)] method for stationary processes.
A general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation. The asymptotic properties of the estimate are derived under the assumption of possible model misspecification. For autoregressive processes with time varying coefficients, the estimate is compared to the least squares estimate. Furthermore, the behavior of estimates is explained when a stationary model is fitted to a nonstationary process.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62F12 Asymptotic properties of parametric estimators
62F10 Point estimation
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[1] AKAIKE, H. 1974. A new look at the statistical model identification. IEEE Trans. Automat. Control AC-19 716 722. Z. · Zbl 0314.62039
[2] AZENCOTT, R. and DACUNHA-CASTILLE, D. 1986. Series of Irregular Observations. Springer, New York. Z. · Zbl 0593.62088
[3] BRILLINGER, D. R. 1981. Time Series: Data Analy sis and Theory. Holden Day, San Francisco. Z.
[4] BROCKWELL, P. and DAVIS, R. A. 1987. Time Series: Theory and Methods. Springer, New York. · Zbl 0604.62083
[5] DAHLHAUS, R. 1983. Spectral analysis with tapered data. J. Time Ser. Anal. 4 163 175. Z. · Zbl 0552.62068
[6] DAHLHAUS, R. 1985. On a spectral density estimate obtained by averaging periodograms. J. Appl. Probab. 22 598 610. Z. JSTOR: · Zbl 0581.62078
[7] DAHLHAUS, R. 1988. Small sample effects in time series analysis: a new asy mptotic theory and a new estimate. Ann. Statist. 16 808 841. Z. · Zbl 0662.62100
[8] DAHLHAUS, R. 1996a. On the Kullback Leibler information divergence of locally stationary processes. Stochastic Process. Appl. 62 139 168. Z. · Zbl 0849.60032
[9] DAHLHAUS, R. 1996b. Maximum likelihood estimation and model selection for locally stationary processes. J. Nonparametric Statist. 6 171 191. Z. · Zbl 0879.62025
[10] DAHLHAUS, R. 1996c. Asy mptotic statistical inference for nonstationary processes with evolutionary spectra. In Athens Conference on Applied Probability and Time Series Analy Z. sis P. M. Robinson and M. Rosenblatt, eds. 2. Springer, New York. Z.
[11] DZHAPARIDZE, K. 1986. Parameter Estimation and Hy pothesis Testing in Spectral Analy sis of Stationary Time Series. Springer, New York. Z. · Zbl 0584.62157
[12] FINDLEY, D. 1985. On the unbiasedness property of AIC for exact or approximating linear stochastic time series models. J. Time Ser. Anal. 6 229 252. Z. · Zbl 0602.62079
[13] GRAy BILL, F. A. 1983. Matrices with Applications in Statistics, 2nd ed. Wadsworth, Belmont, CA.Z. · Zbl 0496.15002
[14] GRENIER, Y. 1983. Time-dependent ARMA modeling of nonstationary signals. IEEE Trans. Acoust. Speech Signal Process. ASSP-31 899 911. Z.
[15] HALLIN, M. 1978. Mixed autoregressive moving-average multivariate processes with timedependent coefficients. J. Multivariate Anal. 8 567 572. Z. · Zbl 0394.62067
[16] HANNAN, E. J. 1973. Multiple Time Series. Wiley, New York. Z. · Zbl 0273.62049
[17] HOSOy A, Y. and TANIGUCHI, M. 1982. A central limit theorem for stationary processes and the parameter estimation of linear processes. Ann. Statist. 10 132 153. Z. · Zbl 0484.62102
[18] KITAGAWA, G. and GERSCH, W. 1985. A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series. IEEE Trans. Automat. Control. AC-30 48 56. Z. · Zbl 0554.62079
[19] KUNSCH, H. R. 1995. A note on causal solutions for locally stationary AR-processes. ETH \" Zurich. Preprint. \" Z.
[20] MELARD, G. and HERTELEER-DE SCHUTTER, A. 1989. Contributions to evolutionary spectral Ň•heory. J. Time Ser. Anal. 10 41 63. Z. · Zbl 0686.62072
[21] MILLER, K. S. 1968. Linear Difference Equations. Benjamin, New York. Z. · Zbl 0162.13403
[22] 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.Z. · Zbl 0871.62081
[23] PRIESTLEY, M. B. 1965. Evolutionary spectra and non-stationary processes. J. Roy. Statist. Soc. Ser. B 27 204 237. Z. JSTOR: · Zbl 0144.41001
[24] PRIESTLEY, M. B. 1981. Spectral Analy sis and Time Series 2. Academic Press, London. Z.
[25] PRIESTLEY, M. B. 1988. Nonlinear and Nonstationary Time Series Analy sis. Academic Press, London. Z.
[26] RIEDEL, K. S. 1993. Optimal data-based kernel estimation of evolutionary spectra. IEEE Trans. Signal Process. 41 2439 2447. Z. · Zbl 0825.93736
[27] SUBBA RAO, T. 1970. The fitting of nonstationary time series models with time-dependent parameters. J. Roy. Statist. Soc. Ser. B 32 312 322. Z. JSTOR: · Zbl 0225.62109
[28] WHITTLE, P. 1953. Estimation and information in stationary time series. Ark. Mat. 2 423 434. Z. · Zbl 0053.41003
[29] YOUNG, P. C. and BEVEN, K. J. 1994. Data-based mechanistic modelling and the rainfall-flow nonlinearity. Environmetrics 5 335 363.
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