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Bayes Methods for Trending Multiple Time Series with an Empirical Application to the US Economy

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Abstract

Multiple time series models with stochastic regressors are considered and primary attention is given to vector autoregressions (VAR's) with trending mechanisms that may be stochastic, deterministic or both. In a Bayesian framework, the data density in such a system implies the existence of a time series "Bayes model" and "Bayes measure" of the data. These are predictive models and measures for the next period observation given the historical trajectory to the present. Issues of model selection, hypothesis testing and forecast evaluation are all studied within the context of these models and the measures are used to develop selection criteria, test statistics and encompassing tests within the compass of the same statistical methodology. Of particular interest in applications are lag order and trend degree, causal effects, the presence and number of unit roots in the system, and for integrated series the presence of cointegration and the rank of the cointegration space, which can be interpreted as an order selection problem. In data where there is evidence of mildly explosive behavior we also wish to allow for the presence of co-motion among variables even though they are individually not modelled as integrated series. The paper develops a statistical framework for addressing these features of trending multiple time series and reports an extended empirical application of the methodology to a model of the US economy that sets out to explain the behavior of and to forecast interest rates, unemployment, money stock, prices and income. The performance of a data-based, evolving "Bayes model" of these series is evaluated against some rival fixed format VAR's, VAR's with Minnesota priors (BVARM's) and univariate models. The empirical results show that fixed format VAR's and BVARM's all perform poorly in forecasting exercises in comparison with evolving "Bayes models" that explicitly adapt in form as new data becomes available.

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File URL: http://cowles.econ.yale.edu/P/cd/d10a/d1025.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1025.

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Length: 69 pages
Date of creation: Aug 1992
Date of revision:
Handle: RePEc:cwl:cwldpp:1025

Note: CFP 914.
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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Bayes model; Bayes measure; causality; cointegration; co-motion; deterministic trend; forecast-encompass; one-period ahead forecasts; order selection; PIC criterion; PICF criterion; RUMPY model; unit root;

References

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  1. Toda, Hiro Y & Phillips, Peter C B, 1993. "Vector Autoregressions and Causality," Econometrica, Econometric Society, vol. 61(6), pages 1367-93, November.
  2. Peter C.B. Phillips & Werner Ploberger, 1992. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection," Cowles Foundation Discussion Papers 1017, Cowles Foundation for Research in Economics, Yale University.
  3. Richard M. Todd, 1990. "Vector autoregression evidence on monetarism: another look at the robustness debate," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Spr, pages 19-37.
  4. Boudjellaba, Hafida & Dufour, Jean-Marie & Roy, Roch, 1994. "Simplified conditions for noncausality between vectors in multivariate ARMA models," Journal of Econometrics, Elsevier, vol. 63(1), pages 271-287, July.
  5. Peter C.B. Phillips, 1990. "To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends," Cowles Foundation Discussion Papers 950, Cowles Foundation for Research in Economics, Yale University.
  6. Zellner, A. & Hong, C., 1988. "Forecasting International Growth Rates Using Bayesian Shrinkage And Other Procedures," Papers m8802, Southern California - Department of Economics.
  7. Poirier, Dale J, 1991. "A Bayesian View of Nominal Money and Real Output through a New Classical Macroeconomic Window," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(2), pages 125-48, April.
  8. Christopher A. Sims & Harald Uhlig, 1988. "Understanding unit rooters: a helicopter tour," Discussion Paper / Institute for Empirical Macroeconomics 4, Federal Reserve Bank of Minneapolis.
  9. Sims, Christopher A, 1972. "Money, Income, and Causality," American Economic Review, American Economic Association, vol. 62(4), pages 540-52, September.
  10. Christopher A. Sims, 1980. "Comparison of Interwar and Postwar Business Cycles: Monetarism Reconsidered," NBER Working Papers 0430, National Bureau of Economic Research, Inc.
  11. Thomas Doan & Robert B. Litterman & Christopher A. Sims, 1986. "Forecasting and conditional projection using realistic prior distribution," Staff Report 93, Federal Reserve Bank of Minneapolis.
  12. Zellner, Arnold, 1985. "Bayesian Econometrics," Econometrica, Econometric Society, vol. 53(2), pages 253-69, March.
  13. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  14. Sims, Christopher A., 1988. "Bayesian skepticism on unit root econometrics," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 463-474.
  15. Feige, Edgar L & Pearce, Douglas K, 1979. "The Casual Causal Relationship between Money and Income: Some Caveats for Time Series Analysis," The Review of Economics and Statistics, MIT Press, vol. 61(4), pages 521-33, November.
  16. Litterman, Robert B, 1986. "Forecasting with Bayesian Vector Autoregressions-Five Years of Experience," Journal of Business & Economic Statistics, American Statistical Association, vol. 4(1), pages 25-38, January.
  17. Osborn, Denise R, 1984. "Causality Testing and Its Implications for Dynamic Econometric Models," Economic Journal, Royal Economic Society, vol. 94(376a), pages 82-96, Supplemen.
  18. Peter C.B. Phillips, 1992. "Bayes Models and Forecasts of Australian Macroeconomic Time Series," Cowles Foundation Discussion Papers 1024, Cowles Foundation for Research in Economics, Yale University.
  19. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-80, November.
  20. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
  21. Fair, Ray C., 1986. "Evaluating the predictive accuracy of models," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 3, chapter 33, pages 1979-1995 Elsevier.
  22. Peter C.B. Phillips, 1992. "Bayesian Model Selection and Prediction with Empirical Applications," Cowles Foundation Discussion Papers 1023, Cowles Foundation for Research in Economics, Yale University.
  23. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-44, January.
  24. Richard M. Todd, 1984. "Improving economic forecasting with Bayesian vector autoregression," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.
  25. Sims, Christopher A, 1980. "Macroeconomics and Reality," Econometrica, Econometric Society, vol. 48(1), pages 1-48, January.
  26. Garcia-Ferrer, Antonio, et al, 1987. "Macroeconomic Forecasting Using Pooled International Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(1), pages 53-67, January.
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  28. Robert B. Litterman, 1984. "Above-average national growth in 1985 and 1986," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.
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Citations

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Cited by:
  1. Peter C.B. Phillips, 1995. "Impulse Response and Forecast Error Variance Asymptotics in Nonstationary VAR's," Cowles Foundation Discussion Papers 1102, Cowles Foundation for Research in Economics, Yale University.
  2. John C. Chao & Peter C.B. Phillips, 1997. "Model Selection in Partially Nonstationary Vector Autoregressive Processes with Reduced Rank Structure," Cowles Foundation Discussion Papers 1155, Cowles Foundation for Research in Economics, Yale University.
  3. Allen, P. Geoffrey & Morzuch, Bernard J., 2006. "Twenty-five years of progress, problems, and conflicting evidence in econometric forecasting. What about the next 25 years?," International Journal of Forecasting, Elsevier, vol. 22(3), pages 475-492.
  4. Phillips, Peter C.B., 2003. "Vision And Influence In Econometrics: John Denis Sargan," Econometric Theory, Cambridge University Press, vol. 19(03), pages 495-511, June.
  5. Phillips, Peter C.B., 2005. "Automated Discovery In Econometrics," Econometric Theory, Cambridge University Press, vol. 21(01), pages 3-20, February.
  6. Peter C.B. Phillips, 2003. "Laws and Limits of Econometrics," Cowles Foundation Discussion Papers 1397, Cowles Foundation for Research in Economics, Yale University.
  7. Phillips, Peter C. B., 1995. "Bayesian prediction a response," Journal of Econometrics, Elsevier, vol. 69(1), pages 351-365, September.
  8. Peter C.B. Phillips, 1995. "Automated Forecasts of Asia-Pacific Economic Activity," Cowles Foundation Discussion Papers 1103, Cowles Foundation for Research in Economics, Yale University.

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