Nonlinear Time Series Modeling and Forecasting for Periodic and ARCH Effects

Abstract

Generalized autoregressive conditional heteroscedastic (GARCH) nonlinear time series model may be employed to describe data sets depicting volatility. This model along with its estimation procedure is thoroughly studied. Lagrange multiplier (LM) test for testing presence of Autoregressive conditional heteroscedastic (ARCH) effects is also discussed. As an illustration, modeling and forecasting of monthly rainfall data of Sub-Himalayan West Bengal meteorological subdivision, India is carried out. As the data exhibits presence of seasonal component, Hylleberg, Engle, Granger and Yoo (1990) [HEGY] seasonal unit root test is applied to the data with a view to make the series stationary through “differencing filter”. Subsequently, GARCH model is employed on the residuals obtained after carrying out Periodic autoregressive (PAR) modeling of the seasonal variation. Further, Mixture periodic ARCH (MPARCH) model, which is an extension of GARCH model, is also applied on zero conditional mean residual series to identify time varying volatility in the data set. The performance of fitted models is examined from the viewpoint of dynamic one-step and two-step ahead forecast error variances along with Mean square prediction error (MSPE), Mean absolute prediction error (MAPE) and Relative mean absolute prediction error (RMAPE). Salient feature of the work done is that, for selected model, best predictor and prediction error variance for carrying out out-of-sample forecasting up to three-steps ahead are derived analytically by recursive use of conditional expectation and conditional variance. The SPSS, SAS and EViews software packages are used for data analysis. By carrying out a comparative study, it is concluded that, for the data under consideration, the PAR model with AR-GARCH errors has performed better than the Seasonal autoregressive integrated moving average (SARIMA) model for modeling as well as forecasting