Time Series Models Of Heteroscedasticity

The models discussed so far concern the conditional mean structure of time series data. However, more recently, there has been much work on modeling the conditional variance structure of time series data—mainly motivated by the needs for financial modeling. Let { _Yt } be a time series of interest. The conditional variance of _Yt given the past Y values, _{Yt − 1},_{Yt − 2},…, measures the uncertainty in the deviation of _Yt from its conditional mean E( _Yt |_{Yt − 1},Yt ? 2,…). If { _Yt } follows some ARIMA model, the (one-stepahead) conditional variance is always equal to the noise variance for any present and past values of the process. Indeed, the constancy of the conditional variance is true for predictions of any fixed number of steps ahead for an ARIMA process. In practice, the (one-step-ahead) conditional variance may vary with the current and past values of the process, and, as such, the conditional variance is itself a random process, often referred to as the conditional variance process. For example, daily returns of stocks are often observed to have larger conditional variance following a period of violent price movement than a relatively stable period. The development of models for the conditional variance process with which we can predict the variability of future values based on current and past data is the main concern of the present chapter. In contrast, the ARIMA models studied in earlier chapters focus on how to predict the conditional mean of future values based on current and past data.

In finance, the conditional variance of the return of a financial asset is often adopted as a measure of the risk of the asset. This is a key component in the mathematical theory of pricing a financial asset and the VaR (Value at Risk) calculations; see, for example, Tsay (2005). In an efficient market, the expected return (conditional mean) should be zero, and hence the return series should be white noise. Such series have the simplest autocorrelation structure. Thus, for ease of exposition, we shall assume in the first few sections of this chapter that the data are returns of some financial asset and are white noise; that is, serially uncorrelated data. By doing so, we can concentrate initially on studying how to model the conditional variance structure of a time series. By the end of the chapter, we discuss some simple schemes for simultaneously modeling the conditional mean and conditional variance structure by combining an ARIMA model with a model of conditional heteroscedasticity.