BAYESIAN MULTIPERIOD FORECASTS FOR ARX MODELS

SHU-ING LIU

Graduate Institute of Statistics, National Central University, Chung-Li, Taiwan 320, R.O.C.

(Received February 21, 1994; revised January 9, 1995)

Abstract. Bayesian multiperiod forecasts for AR models with random independent exogenous variables under normal-gamma and normal-inverted Wishart prior assumptions are investigated. By suitably arranging the integration order of the model’s parameters, a t-density mixture approximation is analytically derived to provide an estimator of the posterior predictive density for any future observation. In particular, a suitable t-density is proposed by a convenient closed form. The precision of the discussed methods is examined by using some simulated data and one set of real data up to lead-six-ahead forecasts. It is found that the numerical results of the discussed methods are rather close. In particular, when sample sizes are sufficiently large, it is encouraging to apply a convenient t-density in practical usage. In fact, this t-density estimator asymptotically converges to the true density.

Key words and phrases: ARX model, Bayesian forecast, t-density mixture, posterior predictive density, random regression.