Conditional heteroscedastic autoregressive moving average models with seasonal patterns

Abstract

The earlier research in time series mainly concentrated on models that assume a constant one-period forecast variance. In reality, however, the assumption may not be met in all cases, especially in economics and finance. Therefore, much recent work has been directed towards the relaxation of the constant conditional variance assumption, namely allowing the conditional variance to change over time and keeping the unconditional variance constant. Tsay (1987) proposed the conditional heteroscedastic autoregressive moving average (CHARMA) model. One of the advantages of the model is that it includes the autoregressive conditional heteroscedastic (ARCH) model and the random coefficient autoregressive (RCA) models as its special cases. Both models characterize time series with varying conditional variance in different representations. Therefore, the CHARMA model is more flexible and is able to model data from a wider perspective. It is also believed that seasonal pattern can be an important phenomenon in the conditional variance and so the purpose of this research is to study seasonal conditional heteroscedasticity and extend the CHARMA model to the seasonal CHARMA model. One of the advantages of our approach is that the relevant time series can be modeled in a parsimonious parameterization. The invertibility and stationarity conditions for the model are derived. We study all the procedures for building up the model. These include the test for varying conditional variance, estimation of the model parameters by the least squares, and the maximum likelihood method and diagnostic checking methodology for testing the adequacy of the fitted model. Two empirical examples are discussed in detail: the exchange rate of US dollar/Japanese Yen and the money supply (M1) of United States. In addition, the ability of capturing volatility will be compared among the proposed model and the GARCH family since the GARCH family is widely used in modeling conditional heteroscedasticity. It is found that the exchange rate money supply have a clear seasonal volatility. The proposed model can capture this effect and produce good forecasts.