fixed format models. For ten out of the thirteen series considered here the Bayes models not only improve on the forecasts of more richly parameterised models but also encompass those forecasts. In effect, the predictive distribution of the best Bayes model explains the forecasts delivered by the rival model. According to our Bayesian forecast-encompassing statistic and given the actual forecast history, the posterior odds favour the Bayes models, sometimes by a factor as high as 30:1, as in the case of the short-term interest rate series.
The models we have considered in this paper are scalar time series models. However, all of the ideas we have employed extend in a natural way to multivariate time series; the statistical theory for this extension will be provided in a subsequent paper. And we hope to conduct some empirical exercises with these multivariate methods on Australian macroeconomic data at a later date.
HANNAN, E. J. and M. DEISTLER ( 1988), The Statistical Theory of Linear Systems, John Wiley and Sons, New York.
HANNAN, E. J. and J. RISSANEN ( 1982), 'Recursive Estimation of ARMA Order', Biometrika, 69, pp. 273-80 [ Corrigenda, Biometrika, 1983, p. 70].
MILLS, J. A. and K. PRASAD ( 1992), "'A Comparison of Model Selection Criteria'", Econometric Reviews, 11, pp. 201-33.
PAULSEN, J. ( 1984), "'Order Determination of Multivariate Autoregressive Time Series with Unit Roots'", Journal of Time Series Analysis, 5, pp. 115-27.
PHILLIPS, P. C. B. ( 1991), "'Bayesian Routes and Unit Roots: de Rebus Prioribus Semper est Disputandum'", Journal of Applied Econometrics, 6( 4), pp. 435-74.
----- ( 1992), "'Bayesian Model Selection and Prediction with Empirical Applications'", mimeo, Yale University.
PHILLIPS, P. C. B. and W. PLOBERGER ( 1991), 'Time Series Modeling with a Bayesian Frame of Reference: I. Concepts and Illustrations', Discussion Paper No. 980, Cowles Foundation.
----- ( 1992), "'Posterior Odds Testing for a Unit Root with Data-based Model Selection'", Discussion Paper No. 1017, Cowles Foundation, to appear in Econometric Theory, 1994.
PÖTSCHER, B. M. ( 1989), "'Model Selection Under Nonstationarity: Autoregressive Models and Stochastic Linear Regression Models'", Annals of Statistics, 17, pp. 1257-74.
RISSANEN, J. ( 1986), "'Stochastic Complexity and Modeling'", Annals of Statistics, 14, pp. 1080-100.
----- ( 1987a), "'Stochastic Complexity'", Journal of the Royal Statistical Society, B, pp. 223-39 and 252-65.
----- ( 1987b), "'Stochastic Complexity and the MDL Principle'", Econometric Reviews, pp. 85-102.
SCHWARZ, G. ( 1978), "'Estimating the Dimension of a Model'", Annals of Statistics, 6, pp. 461-64.
TSAY, R. S. ( 1984), "'Order Selection in Nonstationary Autoregressive Models'", Annals of Statistics, 12, pp. 1425-33.
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Publication information: Book title: Nonstationary Time Series Analysis and Cointegration. Contributors: Colin P. Hargreaves - Editor. Publisher: Oxford University. Place of publication: Oxford. Publication year: 1994. Page number: 86.