Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis

There are several ways to approach Hermite polynomial systems, cumulants and their relationship. Our treatment starts with a general method of calculating the expectation of nonlinear function of Gaussian random variables, see [13] and [127]. Then we define the classical Hermite polynomials and their generalization with several variables. A rather simple introduction to cumulants is given. The diagram formulae are used to show the basic connections between cumulants and Hermite polynomials. These facts will be important for the multiple Wiener-Itô stochastic integrals in the next Chapter. Some general well known theory of stationary processes as spectral representation and higher order spectra are also considered. An approximation of the spectrum and the bispectrum of some nonlinear function of a Gaussian stationary process closes the Chapter.

A basic tool of our investigations of nonlinear problems of time series analysis is the multiple Wiener-Itô integral. In the time domain Wiener started to investigate the stationary functionals of the Brownian motion processes in terms of higher order stochastic integrals. He developed the so called chaotic series representations [146]. The frequency domain analysis of stationary flows in the space of £² functionals of standard Wiener processes came into the picture when Rosenblatt’s noncentral limit problem was studied in more general circumstances by Sinai [115], Dobrushin[33], Dobrushin and Major [34]. See in Major [78] for details. We give here some alternatives for the definition of multiple Wiener-Itô integral for a better understanding of this technique. The chaotic representation of a stationary subordinated processes will be considered as the generalization of the classic spectral representation of stationary random processes. Two particular cases are studied in detail. One is the closest possible process to the Gaussian stationary processes which has only first, i.e., Gaussian and second order multiple terms. The second is an appropriate function of a Gaussian stationary process.

Over the past fifteen years or so a great deal of attention has been paid to a particular class of nonlinear systems, namely to the bilinear ones. One can mention the lecture notes of Subba Rao and Gabr [122] and the paper of Tuan [96] by time series side and the works [4] and [88] by system theory side. Papers discussing bilinear time series include: [2], [22], [30], [40], [47], [46], [45], [51], [62], [64], [69], [76], [75], [81], [83], [88], [95], [102], [103], [105], [101], [98], [99], [112], [77], [113], [111], [117], [116], [119], [120], [62], [126], [139], [128], [132], [141] and [124].

Recently, considerable attention has been paid to nonlinear models in time series analysis. The fact is, most of the time series in practice are not Gaussian, and the second order statistics of a bilinear model, for example, does not contain any information about the parameters of nonlinearity, see Terdik and Subba Rao [139]. The methods of parameter estimation are usually based on either the covariances as Yule-Walker equations or the spectrum, see the monographs by Brockwell and Davis [29], Priestley [100], Rosenblatt [107]. These are called Gaussian estimates because they make use of the second order information only. Brillinger [21] started to apply a criterion involving a third order spectrum, i.e., a bispectrum as well as a second order one and found improvement in estimates. The idea is that the theoretical and the estimated spectra are compared by an iteratively reweighted least squares procedure. The idea of Gaussian estimation in the second order case goes back essentially to the method suggested by Whittle [144]. The properties of such types of estimates are studied by Rice [106], using the asymptotic properties of the spectral estimators due to Brillinger and Rosenblatt [27], and discussed by several authors such as Walker [142], [143], [35] and Hannan [50]. The handicap of the nonGaussian parameter estimation is that the exact spectral and bispectral densities are supposed to be known up to some parameters of the model. The models that have been successfully considered are the linear (nonGaussian) Brillinger [21] and the bilinear one Terdik [131].

A stationary Gaussian time series has the following properties: (i) the residual series of the moving average representation is a sequence of independent (and Gaussian) series and (ii) the best predictor, i.e., the conditional expectation of the observation according to the past is linear. Both properties lead to the notion of the linearity of a time series. We follow Hannan’s [52] definition that the model is linear if the linear predictor is optimal. This assumption seems to be the minimum requirement. This means that the residual sequence e_t fulfils the following conditions:

We have used the linearity test for real data. The linear residual process was calculated by fitting an AR model using the LS method. The order P of the autoregression was determined by AIC criteria.