https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1980.tb01126.x

Threshold Autoregression, Limit Cycles and Cyclical Data

The conventional approach to parametric model fitting of time series is realized through the comparison of various competing models by some ad hoc criterion. Since each of the models is usually specified by the parameters determined by the information from the data, the extension of the classical concept of likelihood to this situation is not obvious. By asking the log likelihood of a model to be an unbiased estimate of the expected log likelihood of the model, a reasonable definition of the likelihood is obtained and this allows us to develop a systematic approach to parametric time series modelling. Practical utility of this approach is demonstrated by numerical examples.

On the Likelihood of a Time Series Model

The article contains an overview over locally stationary processes. At the beginning, time varying autoregressive processes are discussed in detail – both as a deep example and an important class of locally stationary processes. In the next section, a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then, the paper focuses on linear processes where a more general theory is possible. First, a general definition for linear processes is given and time varying spectral densities are discussed in detail. Then, the Gaussian likelihood theory is presented for locally stationary processes. In the next section, the relevance of empirical spectral processes for locally stationary time series is discussed. Empirical spectral processes play a major role in proving theoretical results and provide a deeper understanding of many techniques. The article concludes with an overview of other results for locally stationary processes.

Locally Stationary Processes

A computationally efficient procedure was developed for the fitting of many multivariate locally stationary autoregressive models. The details of the Householder method for fitting multivariate autoregressive model and multivariate locally stationary autoregressive model (MLSAR model) are shown. The proposed procedure is quite efficient in both accuracy and computation. The amount of computation is bounded by a multiple of Nm
2 with N being the data length and m the highest model order, and does not depend on the number of models checked. This facilitates the precise estimation of the change point of the AR model. Based on the AICs' of the fitted MLSAR models and Akaike's definition of the likelihood of the models, a method of evaluating the posterior distribution of the change point of the AR model is also presented. The proposed procedure is, in particular, useful for the estimation of the arrival time of the S wave of a microearthquake. To illustrate the usefulness of the proposed procedure, the seismograms of the foreshocks of the 1982 Urakawa-Oki Earthquake were analyzed. These data sets have been registered to AISM Data Library and the readers of this Journal can access to them by the method described in this issue.

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Estimation of the arrival times of seismic waves by multivariate time series model

Publisher Summary This chapter discusses the use of the minimum akaike information criterion estimation (MAICE) procedure and its conceptual generalization, the entropy maximization principle, in relation to the problem of stochastic system identification. The most important contribution of MAICE is the clarification of the importance of modeling. The systematic approach to the parameter and structure estimation realized by MAICE makes it almost unnecessary to spend researchers' time for the search of ad hoc procedures. The explicit use of likelihood in MAICE makes it possible to provide clear-cut answers to problems which so far have been treated rather unsystematically. A real system may have various possible choices for its model. A Gaussian AR model can approximate a stationary Gaussian process arbitrarily closely by increasing the order. As the maximum likelihood computation for this model reduces, at least approximately, to the least squares computation, this explains why the so-called least squares method can generally be useful for the identification of stochastic systems.

T Ozaki

Papers

On fitting of non-stationary autoregressive models in time series analysis