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On the Shape of the Likelihood/Posterior in Cointegration Models

Published online by Cambridge University Press:  11 February 2009

Frank Kleibergen  and
Herman K. van Dijk
Frank Kleibergen
Affiliation:
Econometric Institute and Tinbergen Institüt
Herman K. van Dijk
Affiliation:
Erasmus University Rotterdam
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Abstract

A vector autoregressive (VAR) model is specified with equation system parameters, which directly reflect the possible cointegrating nature of the analyzed time series. By using a flat/diffuse prior, we show that the marginal posteriors of the parameters of interest (multipliers of the cointegrating vectors) may be nonintegrable and favor difference stationary models in an undesired way. To choose between stationary, cointegrated, and difference stationary models in a meaningful way, the Jeffreys prior principle is used. We investigate the sensitivity of the posterior results with respect to the construction of the Jeffreys prior. In this context, we also analyze the effect of fixed and stochastic initial values. The theoretical results are illustrated by using a VAR model for shortand long–term interest rates in the United States.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 514 - 551
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCE

1.Box, G.E.P. & Tiao, G.C.. A canonical analysis of multiple time series. Biometrika 64 (1977): 355–365.CrossRefGoogle Scholar
2.Campbell, J. & Shiller, R.J.. Cointegration and tests of present value models. Journal of Political Economy 95 (1987): 1062–1088.CrossRefGoogle Scholar
3.DeJong, D.N.Co-integration and trend-stationarity in macro-economic time series: Evidence from the likelihood function. Journal of Econometrics, forthcoming.Google Scholar
4.Drèze, J.H.Bayesian regression using poly t densities. Journal of Econometrics 6 (1977): 329–354.CrossRefGoogle Scholar
5.Drèze, J.H. & Richard, J.F.. Bayesian analysis of simultaneous equations systems. In Griliches, Z. & Intrilligator, M.D. (eds.), Handbook of Econometrics, Vol. 1. Amsterdam: North-Holland, 1983.Google Scholar
6.Engle, R.F. & Granger, C.W.J.. Co-integration and error correction: Representation, estimation and testing. Econometrica 55 (1987): 251–276.CrossRefGoogle Scholar
7.Geweke, J.Bayesian inference in econometric models using Monte Carlo integration. Econometrica 57 (1989): 1317–1339.CrossRefGoogle Scholar
8.Geweke, J.Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics 40 (1989): 63–86.CrossRefGoogle Scholar
9.Hop, J.P. & van Dijk, H.K.. SISAM and MIXIN: Two algorithms for the computation of posterior moments and densities using Monte-Carlo integration. Computer Science in Economics and Management 5 (1992): 183–220.CrossRefGoogle Scholar
10.Johansen, S.Estimation and hypothesis testing of cointegrating vectors in gaussian vector autoregressive models. Econometrica 59 (1991): 1551–1581.CrossRefGoogle Scholar
11.Johansen, J. & Juselius, K.. Maximum likelihood estimation and inference on cointegration – With applications to the demand for money. Oxford Bulletin of Economics and Statistics 52 (1990): 169–210.CrossRefGoogle Scholar
12.Kleibergen, F. & van Dijk, H.K.. Bayesian Simultaneous Equation Model Analysis: On the Existence of Structural Posterior Moments. Econometric Institute report 9269/A, Erasmus University, Rotterdam, 1992.Google Scholar
13.Kleibergen, F. & van Dijk, H.K.. Direct cointegration testing in error correction models. Journal of Econometrics, forthcoming.Google Scholar
14.Kloek, T. & van Dijk, H.K.. Bayesian estimates of equation system parameters: An application of integration by Monte Carlo. Econometrica 46 (1978): 1–20.CrossRefGoogle Scholar
15.Lütkepohl, H.Introduction to Multiple Time Series Analysis. Berlin: Springer-Verlag, 1991.CrossRefGoogle Scholar
16.Magnus, J. & Neudecker, H.. Matrix Differential Calculus with Applications in Statistics and Economics. Chichester: Wiley, 1988.Google Scholar
17.Phillips, P.C.B.Partially identified econometric models. Econometric Theory 5 (1989): 181–240.CrossRefGoogle Scholar
18.Phillips, P.C.B.Optimal inference in cointegrated systems. Econometrica 59 (1991): 283–307.CrossRefGoogle Scholar
19.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
20.Phillips, P.C.B.Bayes Methods for Trending Multiple Time Series with an Empirical Application to the U.S. Economy. Cowles Foundation discussion paper #1025, 1992.Google Scholar
21.Phillips, P.C.B.Some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models. Econometrica 62 (1994): 73–93.CrossRefGoogle Scholar
22.Schotman, P. & van Dijk, H.K.. A Bayesian analysis of the unit root hypothesis. Journal of Econometrics 49 (1991): 195–238.CrossRefGoogle Scholar
23.Stock, J.H. & Watson, M.W.. Testing for common trends. Journal of the American Statistical Association 83 (1988): 1097–1107.CrossRefGoogle Scholar
24.van Dijk, H.K. & Kloek, T.. Further experience in Bayesian analysis using Monte-Carlo integration. Journal of Econometrics 14 (1980): 307–328.CrossRefGoogle Scholar
25.Zellner, A.An Introduction to Bayesian Inference in Econometrics. New York: Wiley, 1971.Google Scholar