Skip to main content Accessibility help
Hostname: page-component-5c569c448b-nqqt6 Total loading time: 0.309 Render date: 2022-07-03T02:53:43.536Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true


Published online by Cambridge University Press:  31 March 2005

Mattias Villani
Sveriges Riksbank and Stockholm University


A Bayesian reference analysis of the cointegrated vector autoregression is presented based on a new prior distribution. Among other properties, it is shown that this prior distribution distributes its probability mass uniformly over all cointegration spaces for a given cointegration rank and is invariant to the choice of normalizing variables for the cointegration vectors. Several methods for computing the posterior distribution of the number of cointegrating relations and distribution of the model parameters for a given number of relations are proposed, including an efficient Gibbs sampling approach where all inferences are determined from the same posterior sample. Simulated data are used to illustrate the procedures and for discussing the well-known issue of local nonidentification.The author thanks Luc Bauwens, Anant Kshirsagar, Peter Phillips, Herman van Dijk, four anonymous referees, and especially Daniel Thorburn for helpful comments. Financial support from the Swedish Council of Research in Humanities and Social Sciences (HSFR) grant F0582/1999 and Swedish Research Council (Vetenskapsrådet) grant 412-2002-1007 is gratefully acknowledged. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank.

Research Article
© 2005 Cambridge University Press

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.)


Ahn, S.K. & G.C. Reinsel (1990) Estimation for partially non-stationary multivariate autoregressive processes. Journal of the American Statistical Association 85, 813823.Google Scholar
Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control AC-19, 716723.Google Scholar
Bauwens, L. & P. Giot (1998) A Gibbs sampler approach to cointegration. Computational Statistics 13, 339368.Google Scholar
Bauwens, L. & M. Lubrano (1996) Identification restrictions and posterior densities in cointegrated Gaussian VAR systems. In T.B. Fomby & R.C. Hill (eds.), Advances in Econometrics, vol. 11, part B, pp. 328. JAI Press.
Bauwens, L., M. Lubrano, & J.-F. Richard (1999) Bayesian Inference in Dynamic Econometric Models. Oxford University Press.
Bauwens, L. & J.-F. Richard (1985) A 1-1 poly-t random variable generator with application to Monte Carlo integration. Journal of Econometrics 29, 1946.Google Scholar
Bauwens, L. & H.K. van Dijk (1990) Bayesian limited information analysis revisited. In Gabszewicz, J.J., Richard, J.-F., & Wolsey, L. (eds.), Economic Decision-Making: Games, Econometrics and Optimisation, pp. 385424. North-Holland.
Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer-Verlag.
Box, G.E.P. & G.C. Tiao (1973) Bayesian Inference in Statistical Analysis. Addison-Wesley.
Chao, J.C. & P.C.B. Phillips (1999) Model selection in partially nonstationary vector autoregressive processes with reduced rank structure. Journal of Econometrics 91, 227271.Google Scholar
Chib, S. (1995) Marginal likelihood from the Gibbs output. Journal of the American Statistical Association 90, 13131321.Google Scholar
Corander, J. & M. Villani (2004) Bayesian assessment of dimensionality in reduced rank regression. Statistica Neerlandica 58, 255270.Google Scholar
Dickey, J.M. (1967) Matric-variate generalizations of the multivariate t distribution and the inverted multivariate t distribution. Annals of Mathematical Statistics 38, 511518.Google Scholar
Dickey, J.M. (1968) Three multidimensional integral identities with Bayesian applications. Annals of Mathematical Statistics 39, 16151627.Google Scholar
Drèze, J.H. (1977) Bayesian regression analysis using poly-t densities. Journal of Econometrics 6, 329354.Google Scholar
Drèze, J.H. & J.-F. Richard (1983) Bayesian analysis of simultaneous equation systems. In Z. Griliches & M.D. Intriligator (eds.), Handbook of Econometrics, vol. 1.
Doan, T., R.B. Litterman, & C.A. Sims (1984) Forecasting and conditional projection using realistic prior distributions. Econometrics Reviews 3, 1100.Google Scholar
Engle, R.F. & C.W.J. Granger (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251276.Google Scholar
Geweke, J. (1989) Bayesian inference in econometric models using Monte Carlo integration. Econometrica 57, 13171340.Google Scholar
Geweke, J. (1996) Bayesian reduced rank regression in econometrics. Journal of Econometrics 75, 121146.Google Scholar
Hannan, E.J. & B.J. Quinn (1979) The determination of the order of an autoregression. Journal of the Royal Statistical Society, Series B 41, 190195.Google Scholar
Harville, D.A. (1997) Matrix Algebra from a Statistician's Perspective. Springer-Verlag.
James, A.T. (1954) Normal multivariate analysis and the orthogonal group. Annals of Mathematical Statistics 25, 4074.Google Scholar
Jeffreys, H. (1961) Theory of Probability, 3rd ed. Oxford University Press.
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 15511580.Google Scholar
Johansen, S. (1995) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press.
Kleibergen, F. & R. Paap (2002) Priors, posteriors and Bayes factors for a Bayesian analysis of cointegration. Journal of Econometrics 111, 223249.Google Scholar
Kleibergen, F. & H.K. van Dijk (1994) On the shape of the likelihood/posterior in cointegration models. Econometric Theory 10, 514551.Google Scholar
Kloek, T. & H.K. van Dijk (1978) Bayesian estimates of equation system parameters: An application of integration by Monte Carlo. Econometrica 46, 119.Google Scholar
Litterman, R.B. (1986) Forecasting with Bayesian vector autoregressions—Five years of experience. Journal of Business & Economic Statistics 4, 2538.Google Scholar
Luukkonen, R., A. Ripatti, & P. Saikkonen (1999) Testing for a valid normalization of cointegration vectors in vector autoregressive processes. Journal of Business & Economic Statistics 17, 195204.Google Scholar
Mardia, K.V. & P.E. Jupp (2000) Directional Statistics. Wiley.
Phillips, P.C.B. (1989) Spherical matrix distributions and Cauchy quotients. Statistics and Probability Letters 8, 5153.Google Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283306.Google Scholar
Phillips, P.C.B. (1994) Some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models. Econometrica 62, 7393.Google Scholar
Phillips, P.C.B. (1996) Econometric model determination. Econometrica 64, 763812.Google Scholar
Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461464.Google Scholar
Smith, A.F.M. & G.O. Roberts (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B 55, 324.Google Scholar
Stock, J.H. & M.W. Watson (1988) Testing for common trends. Journal of the American Statistical Association 83, 10971107.Google Scholar
Strachan, R.W. (2003) Valid Bayesian estimation of the cointegrating error correction model. Journal of Business & Economic Statistics 21, 185195.Google Scholar
Tierney, L. (1994) Markov chains for exploring posterior distributions (with discussion). Annals of Statistics 22, 17011762.Google Scholar
Villani, M. (2000) Aspects of Bayesian Cointegration. Ph.D. thesis, Stockholm University, Sweden.
Villani, M. (2001a) Fractional Bayesian lag length inference in multivariate autoregressive processes. Journal of Time Series Analysis 22, 6786.Google Scholar
Villani, M. (2001b) Bayesian prediction with cointegrated vector autoregressions. International Journal of Forecasting 17, 585605.Google Scholar
Villani, M. (2001c) Bayesian Reference Analysis of Cointegration. Research report 2001:1, Department of Statistics, Stockholm University, Sweden. Available at
Zellner, A. (1971) An Introduction to Bayesian Inference in Econometrics. Wiley.
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *