On Mixture Nonlinear Time-Series Modelling and Forecasting for Arch Effects

Abstract

In the class of Nonlinear time-series models, Gaussian mixture transition distribution (GMTD) and Mixture autoregressive (MAR) models may be employed to describe those data sets that depict sudden bursts, outliers and flat stretches at irregular time-epochs. In order to capture volatility explicitly, recently a new family, viz. MAR-Autoregressive conditional heteroscedastic (MAR-ARCH) has been introduced in the literature. In this paper, these three families are studied by considering weekly wholesale onion price data during April, 1998 to March, 2002. Presence of ARCH in detrended and deseasonalised series is tested by Naive-Lagrange multiplier (Naive-LM) test. Estimation of parameters is done using Expectation-Maximization (EM) algorithm and best model from each family is selected on basis of Bayesian information criterion (BIC). The salient feature of work done is that, for selected models, formulae for carrying out out-of-sample forecasting up to three-steps ahead have been obtained theoretically, perhaps for the first time, by recursive use of conditional expectation and conditional variance. In respect of out-of-sample data, results derived enable us to compute best predictor, prediction error variance, and predictive density. It is concluded that a two-component MAR-ARCH provides best description of the data for modelling as well as forecasting purposes.