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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.

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 |

### Keywords:

model selection; evolutionary spectra; time series; stationarity; ARMA models; nonstationary processes; general minimum distance estimation procedure; misspecification; least squares estimate
Full Text:
DOI

### References:

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