Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-23T23:48:50.793Z Has data issue: false hasContentIssue false
  • English
  • Français

Priors for Macroeconomic Time Series and Their Application

Published online by Cambridge University Press:  11 February 2009

John Geweke
John Geweke
Affiliation:
University of Minnesota and Federal Reserve Bank of Minneapolis
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper takes up Bayesian inference in a general trend stationary model for macroeconomic time series with independent Student-t disturbances. The model is linear in the data, but nonlinear in parameters. An informative but nonconjugate family of prior distributions for the parameters is introduced, indexed by a single parameter that can be readily elicited. The main technical contribution is the construction of posterior moments, densities, and odd ratios by using a six-step Gibbs sampler. Mappings from the index parameter of the family of prior distribution to posterior moments, densities, and odds ratios are developed for several of the Nelson–Plosser time series. These mappings show that the posterior distribution is not even approximately Gaussian, and they indicate the sensitivity of the posterior odds ratio in favor of difference stationarity to the choice of the prior distribution.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 609 - 632
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCE

1.De Finetti, B. The Bayesian approach to the rejection of outliers. Proceedings of the Fourth Berkeley Symposium on Mathematical Probability and Statistics, Vol. 1, pp. 199–210. Berkeley: University of California Press, 1961.Google Scholar
2.DeJong, D.N. & Whiteman, C.H.. Reconsidering “Trends and random walks in macroeconomic time series.” Journal of Monetary Economics 28 (1991): 221–254.CrossRefGoogle Scholar
3.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
4.Doan, T., Litterman, R. & Sims, C.A.. Forecasting and conditional projection using realistic prior distributions. Econometric Reviews 3(1) (1984): 1–56.CrossRefGoogle Scholar
5.Fraser, D.A.S.Necessary analysis and adaptive inference (with discussion). Journal of the American Statistical Association 71 (1976): 99–113.CrossRefGoogle Scholar
6.Fraser, D.A.S.Inference and Linear Models. New York: McGraw-Hill, 1979.Google Scholar
7.Gelfand, A.E. & Smith, A.F.M.. Sampling based approaches to calculating marginal densities (with discussion). Journal of the American Statistical Association 85 (1990): 398–409.CrossRefGoogle Scholar
8.Geman, S. & Geman, D.. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6 (1984): 721–741.CrossRefGoogle ScholarPubMed
9.Geweke, J. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Berger, J.O., Bernardo, J.M., Dawid, A.P. & Smith, A.F.M. (eds.), Bayesian Statistics 4, pp. 169–194. Oxford: Oxford University Press, 1992.Google Scholar
10.Geweke, J.Priors for Macroeconomic Time Series and Their Application. Minneapolis: Federal Reserve Bank of Minneapolis, Institute for Empirical Macroeconomics discussion paper #64, 1992.CrossRefGoogle Scholar
11.Geweke, J.Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics 8 (1993): 519–540.CrossRefGoogle Scholar
12.Hannan, E.J.Multiple Time Series. New York: Wiley, 1970.CrossRefGoogle Scholar
13.Harrison, P.J. & Stevens, C.F.. Bayesian forecasting (with discussion). Journal of the Royal Statistical Society Series B 38 (1976): 205–247.Google Scholar
14.IMSL. IMSL Math/Library User's Manual: FORTRAN Subroutines for Mathematical Applications. Houston: Author, 1989.Google Scholar
15.Jeffreys, H.Theory of Probability. Oxford: Clarendon Press, 1939.Google Scholar
16.Lange, K.L., Little, R.J.A. & Taylor, J.M.G.. Robust statistical modeling using the t distribution. Journal of the American Statistical Association 84 (1989): 881–896.Google Scholar
17.Leonard, T.A Bayesian approach to the linear model with unequal variances. Technometrics 17 (1975): 95–102.CrossRefGoogle Scholar
18.Maronna, R.A.Robust M-estimators of multivariate location and scatter. Annals of Statistics 4 (1976): 51–67.CrossRefGoogle Scholar
19.Nelson, C.R. & Plosser, C.I.. Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10 (1982): 139–162.CrossRefGoogle Scholar
20.Nummelin, E.General Irreducible Markov Chains and Non-Negative Operators. Cambridge: Cambridge University Press, 1984.CrossRefGoogle Scholar
21.Osiewalski, J. & Steel, M.F.J.. Robust Bayesian inference in elliptical regression models. Journal of Econometrics 57 (1993): 345–363.CrossRefGoogle Scholar
22.Perron, P.The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57 (1989): 1361–1401.CrossRefGoogle Scholar
23.Phillips, P.C.B.Time series regressions with a unit root. Econometrica 55 (1987): 277–301.CrossRefGoogle Scholar
24.Phillips, P.C.B.To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6 (1991): 333–364.CrossRefGoogle Scholar
25.Phillips, P.C.B.Bayesian routes and unit roots: De rebus prioribus semper est disputandum. Journal of Applied Econometrics 6 (1991): 435–473.CrossRefGoogle Scholar
26.Ramsay, J.O. & Novick, M.R.. PLU robust Bayesian decision theory: Point estimation. Journal of the American Statistical Association 75 (1980): 901–907.CrossRefGoogle Scholar
27.Revuz, D.Markov Chains. Amsterdam: North-Holland, 1975.Google Scholar
28.Said, S.E. & Dickey, D.A.. Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71 (1984): 599–607.CrossRefGoogle Scholar
29.Schotman, P.C. & van Dijk, H.K.. A Bayesian analysis of the unit root in real exchange rates. Journal of Econometrics 49 (1991): 195–238.CrossRefGoogle Scholar
30.Schotman, P.C. & van Dijk, H.K.. On Bayesian routes to unit roots. Journal of Applied Econometrics 6 (1991): 387–401.CrossRefGoogle Scholar
31.Schotman, P.C. & van Dijk, H.K.. Posterior analysis of possibly integrated time series with an application to real GNP. In Brillinger, D. et al. (eds.), New Directions in Time Series Analysis, Part II, pp. 341–362. Berlin: Springer-Verlag, 1992.Google Scholar
32.Sims, C.A.Bayesian skepticism on unit root econometrics. Journal of Economic Dynamics and Control 12 (1988): 436–474.CrossRefGoogle Scholar
33.Sims, C.A. Comment by Christopher A. Sims on “To criticize the critics,” by Peter C.B. Phillips. Journal of Applied Econometrics 6 (1991): 423–434.CrossRefGoogle Scholar
34.Sims, C.A., Stock, J.H. & Watson, M.W.. Inference in linear time series models with some unit roots. Econometrica 58 (1990): 113–144.CrossRefGoogle Scholar
35.Sims, C.A. & Uhlig, H.. Understanding unit rooters: A helicopter tour. Econometrica 59 (1991): 1591–1599.CrossRefGoogle Scholar
36.Tierney, L.Markov Chains for Exploring Posterior Distributions. University of Minnesota School of Statistics technical report #560, 1991.Google Scholar
37.West, M.Outlier models and prior distributions in Bayesian linear regression. Journal of the Royal Statistical Society Series B 46 (1984): 431–439.Google Scholar
38.Zellner, A.Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. Journal of the American Statistical Association 71 (1976): 400–405.Google Scholar
39.Zellner, A. & Tiao, G.C.. Bayesian analysis of the regression model with autocorrelated errors. Journal of the American Statistical Association 59 (1964): 763–778.CrossRefGoogle Scholar