ON DOUBLE-THRESHOLD AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTIC MODELS WITH APPLICATIONS

Abstract

The double-threshold autoregressive conditional heteroscedastic (DT ARCH) model is motivated by the recent advances in nonlinear time series analysis and the need to model volatility in economic and financial applications. The model structure of the DTARCH model is an extension of Tong's popular and useful threshold model and extends Tong's model to include piecewise linear specifications of the conditional variance. This exercise provides an expository account of how the DTARCH models is used to model the possible asymmetry in both the mean and the variance of the financial series (monthly S&P's Index returns) and the macroeconomic data series (quarterly UK exports growth rate series). The presence of asymmetry in the data series suggests that the conventional linear time series modeling should be scrutinized. It hints at a form of nonlinearity manifested by some asymmetry in the data series. The DTARCH model is thus tentatively used to model these data series since it is capable of capturing asymmetric behavior in the mean and the variance that could be present in financial series and economic time series. The DTARCH model provides a reasonable fit to both the financial series and the macroeconomic time series. Applications of the DTARCH model is still at its infancy stage, and many questions about the usefulness of the model remain unanswered. Until now, in the empirical literature, the empirical result that has been reported so far is in the modeling of financial data. Clearly, the DTARCH model with threshold structures in both the mean and the variance should have many potential applications. The successful application of the DTARCH model to the UK exports growth rate series indicates that this model is a potential class of nonlinear models for modeling of economic time series. Furthermore, out-of-sample forecasts may be examined to compare the performance of the DTARCH models with linear models as well as other nonlinear models. Unfortunately, this aspect of forecasting using DTARCH model is still uncovered.