An Empirical Bayesian Approach to Cointegrating Rank Selection and Test of the Present Value Model for Stock Prices

  • John C. Chao
  • Peter C. B. Phillips


This paper provides an empirical Bayesian approach to the problem of jointly estimating the lag order and the cointegrating rank of a partially non-stationary reduced rank regression. The method employed is a variant of the Posterior Information Criterion (PIC) of Phillips and Ploberger (1994, 1995) and is similar to the asymptotic predictive odds version of the PIC criterion given in Phillips (1994). Here, we use a proper (Gaussian) prior whose hyperparameters are estimated from an initial subsample of the data. The form of the prior is suggested by the asymptotic posterior distribution of the parameters of the model, and, hence, the criterion can be interpreted as an approximate predictive odds ratio in the case where the sample size is large. Applying this procedure to the extended Campbell-Shiller data set for stock prices and dividends, we find the present value model for stock prices to be inconsistent with the data.


Unit Root Stock Price Unit Root Test Stock Prex Prior Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahn, S. K. and G. C. Reinsel (1990). “Estimation for Partially Nonstationary Multivariate Autoregressive Models,” Journal of the American Statistical Association, 85, 813–823CrossRefzbMATHMathSciNetGoogle Scholar
  2. Atkinson, A. C. (1978). “Posterior Probabilities for Choosing a Regression Model,” Biometrika, 65, 39–48CrossRefzbMATHMathSciNetGoogle Scholar
  3. Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-VerlagCrossRefzbMATHGoogle Scholar
  4. Campbell, J. Y. and R. J. Shiller (1987). “Cointegration and Tests of Present Value Models,” Journal of Political Economy, 95, 1062–1088CrossRefGoogle Scholar
  5. Chao, J (1995). “Some Simulation Results of Likelihood Ratio Tests for Cointegrating Ranks in Misspecified Vector Autoregressions,” in preparationGoogle Scholar
  6. Chao, JC and PCB Phillips (1994). “Bayesian Model Selection in Partially Non- stationary Vector Autoregressive Processes with Reduced Rank Structure,” mimeographed, Yale UniversityGoogle Scholar
  7. Cowles, A. (1939). Common Stock Indexes. Bloomington, IN: PrincipiaGoogle Scholar
  8. DeJong, D. N. and C. H. Whiteman (1992). “More Unsettling Evidence on the Perfect Markets Hypothesis: Trend-Stationarity Revisited,” Federal Reserve Bank of Atlanta Economic Review, 77, 1–13Google Scholar
  9. DeJong, D. N. and C. H. Whiteman (1994). “Modeling Stock Prices without Knowing How to Induce Stationary,” Econometric Theory, 10, 701–719CrossRefGoogle Scholar
  10. Engle, R. F. and C. W. J. Granger (1987). “Cointegration and Error-correction: Representation, Estimation, and Testing,” Econometrica, 55, 251–276CrossRefzbMATHMathSciNetGoogle Scholar
  11. Geweke, J. (1994). “Bayesian Comparison of Econometric Models,” Federal Reserve Bank of Minneapolis Research Department working paper 532, July.Google Scholar
  12. Hansen, L. P. and T. Sargent (1981). “Exact Linear Rational Expectations Models: Specification and Estimation,” Staff Report No. 71, Minneapolis: Federal Reserve BankGoogle Scholar
  13. Johansen, S. (1988). “Statistical Analysis of Cointegrating Vectors,” Journal of Economic Dynamic and Control, 12, 231–254CrossRefzbMATHMathSciNetGoogle Scholar
  14. Johansen, S. (1991). “Estimation and Hypothesis Testing of Cointegrating Vectors in Gaussian Vector Autoregressive Models,” Econometrica, 59, 1551–1580CrossRefzbMATHMathSciNetGoogle Scholar
  15. Johansen, S. (1992). “Determination of Cointegrating Rank in the Presence of a Linear Trend,” Oxford Bulletin of Economics and Statistics, 54, 383–397CrossRefGoogle Scholar
  16. Kleidon, A. W. (1986). “Variance Bounds Test and Stock Price Valuation Models,” Journal of Political Economy, 94, 953–1001CrossRefGoogle Scholar
  17. Marsh, T. A. and R. C. Merton (1986). “Dividend Variability and Variance Bounds Tests for the Rationality of Stock Prices,” American Economic Review, 46, 483–498Google Scholar
  18. O’Hagan, A. (1991). “Discussion on Posterior Bayes Factor (by M. Aitkin),” Journal of the Royal Statistical Society Series B, 53, 136Google Scholar
  19. Phillips, P. C. B. (1992). “Bayes Methods for Trending Multiple Time Series with an Empirical Application to the U.S. Economy,” Cowles Foundation Discussion Paper No. 1025, Yale UniversityGoogle Scholar
  20. Phillips, PCB (1994). “Model Determination and Macroeconomic Activity,” mimeographed, Yale UniversityGoogle Scholar
  21. Phillips, PCB (1995). “Bayesian Model Selection and Prediction with Empirical Applications,” Journal of Econometrics(in press)Google Scholar
  22. Phillips, P. C. B. and W. Ploberger (1994). “Posterior Odds Testing for a Unit Root with Data-based Model Selection,” Econometric Theory, 10, 774–808CrossRefMathSciNetGoogle Scholar
  23. Phillips, PCB and W Ploberger (1995). “An Asymptotic Theory of Bayesian Inference for Time Series,” Econometrica(forthcoming)Google Scholar
  24. Phillips, P. C. B. and S. Ouliaris (1988). “Testing for Cointegration Using Principal Components Methods,” Journal of Economic Dynamics and Control, 12, 205–230CrossRefzbMATHMathSciNetGoogle Scholar
  25. Potscher, BM (1983), “Order Estimation in ARM A Models by Lagrange Multiplier Tests,” Annals of Statistics11, 872–885CrossRefMathSciNetGoogle Scholar
  26. Sawa, T. (1978). “Information Criteria for Discriminating among Alternative Regression Models,” Econometrica, 46, 1273–1291CrossRefzbMATHMathSciNetGoogle Scholar
  27. Shibata, R. (1976). “Selection of the Order of an Autoregressive Model by Akaike’s Information Criterion,” Biometrika r63, 117–126CrossRefzbMATHGoogle Scholar
  28. Toda, HY (1991). “An ECM Approach to Tests of Present Value Models,” mimeographed, Yale UniversityGoogle Scholar
  29. Toda, H. Y. and P. C. B. Phillips (1994). “Vector Autoregression and Causality: A Theoretical Overview and Simulation Study,” Econometric Reviews, 13, 259–285CrossRefzbMATHMathSciNetGoogle Scholar
  30. Tsay, R. S. (1984). “OrdeT Selection in Nonstationary Autoregressive Models,” Annals of Statistics, 12, 1425–1433CrossRefzbMATHMathSciNetGoogle Scholar
  31. Wei, C. Z. (1992). “On Predictive Least Squares Principles,” Annals of Statistics, 20, 1–42CrossRefzbMATHMathSciNetGoogle Scholar
  32. West, K. D. (1988). “Dividend Innovations and Stock Market Volatility,” Econometrica, 56, 37–62CrossRefGoogle Scholar
  33. Zellner, A. (1971). An Introduction to Bayesian Inference to Econometrics. New York: John Wiley and SonszbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • John C. Chao
    • 1
  • Peter C. B. Phillips
    • 2
  1. 1.Department of EconomicsUniversity of MarylandCollege ParkUSA
  2. 2.Cowles Foundation for Research in EconomicsYale UniversityNew HavenUSA

Personalised recommendations