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Published online by Cambridge University Press:  12 November 2012

Markku Lanne*
University of Helsinki
Pentti Saikkonen
University of Helsinki
*Address correspondence to Markku Lanne, Department of Political and Economic Studies, University of Helsinki, P.O. Box 17 (Arkadiankatu 7), FIN–00014 University of Helsinki, Finland; e-mail:


In this paper, we propose a new noncausal vector autoregressive (VAR) model for non-Gaussian time series. The assumption of non-Gaussianity is needed for reasons of identifiability. Assuming that the error distribution belongs to a fairly general class of elliptical distributions, we develop an asymptotic theory of maximum likelihood estimation and statistical inference. We argue that allowing for noncausality is of particular importance in economic applications that currently use only conventional causal VAR models. Indeed, if noncausality is incorrectly ignored, the use of a causal VAR model may yield suboptimal forecasts and misleading economic interpretations. Therefore, we propose a procedure for discriminating between causality and noncausality. The methods are illustrated with an application to interest rate data.

Copyright © Cambridge University Press 2012 

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We thank Martin Ellison, Juha Kilponen, Mika Meitz, Antti Ripatti, three anonymous referees, and the co-editor, Robert Taylor, for useful comments. Financial support from the Academy of Finland and the OP-Pohjola Group Research Foundation is gratefully acknowledged. The first version of this paper was completed in May 2009. It was written while the second author worked at the Bank of Finland, whose hospitality is gratefully acknowledged.


Alessi, L., Barigozzi, M., & Capasso, M. (2008) Non-fundamentalness in structural econometric models: A review. International Statistical Review 79, 1647.CrossRefGoogle Scholar
Andrews, B., Davis, R.A., & Breidt, F.J. (2006) Maximum likelihood estimation for all-pass time series models. Journal of Multivariate Analysis 97, 16381659.CrossRefGoogle Scholar
Breidt, J., Davis, R.A., Lii, K.S., & Rosenblatt, M. (1991) Maximum likelihood estimation for noncausal autoregressive processes. Journal of Multivariate Analysis 36, 175198.CrossRefGoogle Scholar
Breidt, J., Davis, R.A., & Trindade, A.A. (2001) Least absolute deviation estimation for all-pass time series models. Annals of Statistics 29, 919946.Google Scholar
Brockwell, P.J. & Davis, R.A. (1987) Time Series: Theory and Methods. Springer-Verlag.CrossRefGoogle Scholar
Campbell, J.Y. & Shiller, R.J. (1991) Yield spreads and interest rate movements: A bird’s eye view. Review of Economic Studies 58, 495514.CrossRefGoogle Scholar
Chan, K.S. & Ho, L. (2004) On the Unique Representation of Non-Gaussian Multivariate Linear Processes. Technical report 341, University of Iowa, Scholar
Chan, K.S., Ho, L., & Tong, H. (2006) A note on time-irreversibility of multivariate linear processes. Biometrika 93, 221227.CrossRefGoogle Scholar
Davis, R.A. & Song, L. (2010) Noncausal Vector AR Processes with Application to Financial Time Series. Technical report, Columbia University.Google Scholar
Duffee, G. (2002) Term premia and interest rate forecasts in affine models. Journal of Finance 57, 405443.CrossRefGoogle Scholar
Fang, K.T., Kotz, S., & Ng, K.W. (1990) Symmetric Multivariate and Related Distributions. Chapman and Hall.CrossRefGoogle Scholar
Francq, C., Roy, R., & Zakoïan, J.-M. (2005) Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100, 532544.CrossRefGoogle Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.CrossRefGoogle Scholar
Johansen, S. & Juselius, K. (2010) An Invariance Property of the Common Trends under Linear Transformations of the Data. CREATES Research papers 2010-72, School of Economics and Management, University of Aarhus.CrossRefGoogle Scholar
Kohn, R. (1979). Asymptotic estimation and hypothesis testing results for vector linear time series models. Econometrica 47, 10051029.CrossRefGoogle Scholar
Lanne, M., Luoto, J., & Saikkonen, P. (2012) Optimal forecasting of noncausal autoregressive time series. International Journal of Forecasting 28, 623631.CrossRefGoogle Scholar
Lanne, M. & Saikkonen, P. (2011) Noncausal autoregressions for economic time series. Journal of Time Series Econometrics 3 (3), Article 2.CrossRefGoogle Scholar
Lof, M. (2012) Noncausality and asset pricing. Studies in Nonlinear Dynamics and Econometrics, forthcoming.Google Scholar
Lütkepohl, H. (1996) Handbook of Matrices. Wiley.Google Scholar
Lütkepohl, H. (2005) New Introduction to Multiple Time Series Analysis. Springer-Verlag.CrossRefGoogle Scholar
Muirhead, R.J. & Waternaux, C.M. (1980) Asymptotic distributions in canonical correlation analysis and other multivariate procedures for nonnormal populations. Biometrika 67, 3143.CrossRefGoogle Scholar
Rosenblatt, M. (2000) Gaussian and Non-Gaussian Linear Time Series and Random Fields. Springer-Verlag.CrossRefGoogle Scholar
Rothenberg, T.J. (1971) Identification in parametric models. Econometrica 39, 577591.CrossRefGoogle Scholar
Sargent, T.J. (1979) A note on maximum likelihood estimation of the rational expectations model of the term structure. Journal of Monetary Economics 5, 133143.CrossRefGoogle Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Cambridge University Press.CrossRefGoogle Scholar
Wong, C.H. & Wang, T. (1992) Moments for elliptically countered random matrices. Sankhy $\bar a$ 54, 265277.Google Scholar
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