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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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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
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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

Related research

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. Richard M. Todd, 1984. "Improving economic forecasting with Bayesian vector autoregression," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.
  2. Boudjellaba, H. & Dufour, J.M. & Roy, R., 1992. "Simplified Conditions for Non-Causality Between Vectors in Multivariate Arma Models," Cahiers de recherche 9236, Universite de Montreal, Departement de sciences economiques.
  3. Phillips, Peter C.B. & Ploberger, Werner, 1994. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 774-808, August.
  4. Sims, Christopher A, 1980. "Macroeconomics and Reality," Econometrica, Econometric Society, vol. 48(1), pages 1-48, January.
  5. Toda, Hiro Y & Phillips, Peter C B, 1993. "Vector Autoregressions and Causality," Econometrica, Econometric Society, vol. 61(6), pages 1367-93, November.
  6. 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.
  7. 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.
  8. Sims, Christopher A & Uhlig, Harald, 1991. "Understanding Unit Rooters: A Helicopter Tour," Econometrica, Econometric Society, vol. 59(6), pages 1591-99, November.
  9. 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.
  10. Christopher A. Sims, 1988. "Bayesian skepticism on unit root econometrics," Discussion Paper / Institute for Empirical Macroeconomics 3, Federal Reserve Bank of Minneapolis.
  11. Garcia-Ferrer, A. & Highfield, R.A. & Palm, F. & Zellner, A., 1987. "Macroeconomic forecasting using pooled international data," Open Access publications from Maastricht University urn:nbn:nl:ui:27-6001, Maastricht University.
  12. Fair, Ray C & Shiller, Robert J, 1990. "Comparing Information in Forecasts from Econometric Models," American Economic Review, American Economic Association, vol. 80(3), pages 375-89, June.
  13. Christopher A. Sims, 1980. "Comparison of Interwar and Postwar Business Cycles: Monetarism Reconsidered," NBER Working Papers 0430, National Bureau of Economic Research, Inc.
  14. 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.
  15. 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.
  16. 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.
  17. Robert B. Litterman, 1984. "Above-average national growth in 1985 and 1986," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall.
  18. Zellner, Arnold, 1985. "Bayesian Econometrics," Econometrica, Econometric Society, vol. 53(2), pages 253-69, March.
  19. Zellner, A. & Hong, C., 1988. "Forecasting International Growth Rates Using Bayesian Shrinkage And Other Procedures," Papers m8802, Southern California - Department of Economics.
  20. 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.
  21. 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.
  22. Sims, Christopher A, 1972. "Money, Income, and Causality," American Economic Review, American Economic Association, vol. 62(4), pages 540-52, September.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
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Citations

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Cited by:
  1. 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.
  2. Peter C.B. Phillips, 2003. "Laws and Limits of Econometrics," Cowles Foundation Discussion Papers 1397, Cowles Foundation for Research in Economics, Yale University.
  3. Phillips, Peter C. B., 1995. "Bayesian prediction a response," Journal of Econometrics, Elsevier, vol. 69(1), pages 351-365, September.
  4. 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.
  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. "Vision and Influence in Econometrics: John Denis Sargan," Cowles Foundation Discussion Papers 1393, Cowles Foundation for Research in Economics, Yale University.
  7. 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.
  8. Phillips, Peter C. B., 1998. "Impulse response and forecast error variance asymptotics in nonstationary VARs," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 21-56.

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