Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 1, 111--129

Posterior

On Mixture Nonlinear Time-series Modelling and Forecasting for ARCH Effects

Himadri Ghosh, M.A. Iquebal and Prajneshu, Indian Agricultural Statistics Research Institute, New Delhi

SUMMARY. 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.

AMS (1991) subject classification. Primary 62P20, 62J02.

Key words and phrases. Autoregressive conditional heteroscedasticity, GMTD model, MAR model, MAR-ARCH model, EM algorithm, volatility, stochastic trend, BIC, out-of-sample forecasting.

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