Conditional Heteroscedastic Time Series Models

Abstract

Under the traditional linear time series or regression setting, the conditional variance of one-step-ahead prediction is time invariant. Experience in conjunction with data analysis, however, suggests that the variability of a process might well depend on the available information. This reality has motivated extensive research to relax the constant variance assumption imposed by the traditional linear time series model, and several classes of generalized parametric models designed specifically for handling nonhomogeneity of a process have been proposed recently. In particular, the random coefficient autoregressive (RCA) models were widely investigated by time series analysts and the autoregressive conditional heteroscedastic (ARCH) models were investigated by econometricians.

The interesting fact is that the ARCH processes can be regarded as special cases of the RCA model. In this article, I first give the relationship between these two types of models and show that the special feature of these two models is the varying conditional variance. This feature is presented in two ways from which the two models differ. The RCA models allow for the conditional variance to evolve with previous observations, whereas the ARCH models make use of previous innovations. This recognition immediately points out that to describe the varying conditional variance the RCA and ARCH models may not be parsimonious in parameterization and further extension is needed. The second goal of this article is, therefore, to propose a general class of conditional heteroscedastic time series models, each of which consists of a constant coefficient autoregressive moving average (ARMA) model and a purely random coefficient transfer function model. Both the RCA and the ARCH models become special cases of this general model. In addition, the proposed model also allows for the conditional variance to be a function of the square of the conditional expectation. The use of a purely random coefficient transfer function model to describe the varying conditional variance is based on the fact that the relationship between the conditional variance and available information must be unidirectional.

Properties of the proposed general models are discussed, and a model specification procedure is suggested. The information criterion approach is adopted to specify the order of the transfer function equation, and simulation is used to illustrate the performance of the criteria, for example, Akaike's information criterion (AIC) and the Bayesian information criterion (BIC). Finally, two applications of the proposed model are considered. The first application uses the model as an alternative for handling outliers in time series analysis. An apparent advantage here is that the model does not require identification of the outliers, for example, location, type, and number of outliers. Series C of Box and Jenkins (1976) provides a good example. The second application of the proposed model is to refine the traditional ARMA technique so that volatility of a series can be adequately addressed. An advantage here is the ability to provide adaptive forecasting intervals. The weekly spot rate of the British pound provides a clear illustration. This example also gives justifications for the model generalization considered in the article.

Key Words:

• CHARMA model
• Extended sample autocorrelation function
• Information criterion
• Random coefficient model
• Transfer function