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Time Series Modeling with a Bayesian Frame of Reference: Concepts, Illustrations and Asymptotics

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Abstract

This paper offers an approach to time series modeling that attempts to reconcile classical and Bayesian methods. The central idea put forward to achieve this reconciliation is that the Bayesian approach relies implicitly on a frame of reference for the data generating mechanism that is quite different from the one that is employed in the classical approach. Differences in inferences from the two approaches are therefore to be expected unless the altered frame of reference is taken into account. We show that the new frame of reference in Bayesian inference is a consequence of a change of measure that arises naturally in the application of Bayes theorem. Our paper explores this change of measure and its consequences using martingale methods. Examples are given to illustrate its practical implications. No assumptions concerning stationarity or rates of convergence are required in the development of our asymptotic theory. Some implications for statistical testing are explored and we suggest new tests, which we call Bayes model tests, for discriminating between models. A posterior odds version of these tests is developed and shown to have good finite sample properties. This is the test that we recommend for practical use. Autoregressive models with multiple lags and deterministic trends are considered and explicit forms are given for the posterior odds tests for the presence of a unit root and for joint tests for the presence of a unit root, drift and trend. This paper emphasizes the new conceptual framework for thinking about Bayesian methods in time series and provides illustrations of its use in some common models for possibly nonstationary time series. A sequel to the present paper develops a general and more abstract theory that will have a wider range of applications.

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File URL: http://cowles.econ.yale.edu/P/cd/d10a/d1038.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 1038.

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Length: 66 pages
Date of creation: Oct 1992
Date of revision:
Handle: RePEc:cwl:cwldpp:1038

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Autoregression; Bayes model; Bayes measure; Bayes test; Bayesian inference; data density process; Deleans exponential; exponential Bayes measure; likelihood; martingale; posterior process; prior density; quadratic variation process; stochastic differential equation; unit root;

References

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  1. Peter C.B. Phillips, 1988. "Optimal Inference in Cointegrated Systems," Cowles Foundation Discussion Papers 866R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
  2. 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.
  3. Schotman, Peter & van Dijk, Herman K., 1991. "A Bayesian analysis of the unit root in real exchange rates," Journal of Econometrics, Elsevier, vol. 49(1-2), pages 195-238.
  4. Sims, Christopher A., 1988. "Bayesian skepticism on unit root econometrics," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 463-474.
  5. Poirier, Dale J, 1988. "Frequentist and Subjectivist Perspectives on the Problems of Model Building in Economics," Journal of Economic Perspectives, American Economic Association, vol. 2(1), pages 121-44, Winter.
  6. Phillips, P.C.B., 1986. "Testing for a Unit Root in Time Series Regression," Cahiers de recherche 8633, Universite de Montreal, Departement de sciences economiques.
  7. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  8. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 2," Cowles Foundation Discussion Papers 819R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1987.
  9. 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.
  10. Kloek, Tuen & van Dijk, Herman K, 1978. "Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo," Econometrica, Econometric Society, vol. 46(1), pages 1-19, January.
  11. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
  12. Peter C.B. Phillips, 1981. "Marginal Densities of Instrumental Variable Estimators in the General Single Equation Case," Cowles Foundation Discussion Papers 609, Cowles Foundation for Research in Economics, Yale University.
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Citations

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Cited by:
  1. Werner Ploberger & Peter C. B. Phillips, 2003. "Empirical Limits for Time Series Econometric Models," Econometrica, Econometric Society, vol. 71(2), pages 627-673, March.
  2. Peter C.B. Phillips, 1994. "Nonstationary Time Series and Cointegration: Recent Books and Themes for the Future," Cowles Foundation Discussion Papers 1081, Cowles Foundation for Research in Economics, Yale University.
  3. Phillips, Peter C B & McFarland, James W & McMahon, Patrick C, 1996. "Robust Tests of Forward Exchange Market Efficiency with Empirical Evidence from the 1920s," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(1), pages 1-22, Jan.-Feb..

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