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Graphical modelling of multivariate time series
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  • Open Access
  • Published: 15 February 2011

Graphical modelling of multivariate time series

  • Michael Eichler1 

Probability Theory and Related Fields volume 153, pages 233–268 (2012)Cite this article

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Abstract

We introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series. The modelling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear dependences. The models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs. In these graphs each component series is represented by a single vertex and directed edges indicate possible Granger-causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure. We introduce various notions of Granger-causal Markov properties and discuss the relationships among them and to other Markov properties that can be applied in this context. Examples for graphical time series models include nonlinear autoregressive models and multivariate ARCH models.

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References

  1. Aalen, O.O.: Dynamic modeling and causality. Scand. Actuar. J. 177–190 (1987)

  2. Andersson S.A., Madigan D., Perlman M.D.: Alternative Markov properties for chain graphs. Scand. J. Stat. 28, 33–85 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold M., Günther R.: Adaptive parameter estimation in multivariate self-exciting threshold autoregressive models. Commun. Stat. Simul. Comput. 30, 257–275 (2001)

    Article  MATH  Google Scholar 

  4. Baba, Y., Engle, R.F., Kraft, D.F. Kroner, K.F.: Multivariate simultaneous generalized ARCH. Tech. rep. Department of Economics, University of California, San Diego (1990)

  5. Baccalá L.A., Sameshima K.: Partial directed coherence: a new concept in neural structure determination. Biol. Cybern. 84, 463–474 (2001)

    Article  MATH  Google Scholar 

  6. Bollerslev T.: Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach. Rev. Econ. Stat. 72, 498–505 (1990)

    Article  Google Scholar 

  7. Bollerslev T., Engle R.F., Nelson D.B.: ARCH models. In: Engle, R.F., Mc Fadden, D.L. (eds) Handbook of Econometrics, vol. IV, Elsevier, Amsterdam (1994)

    Google Scholar 

  8. Brillinger D.R.: Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59, 189–200 (1988)

    Article  MATH  Google Scholar 

  9. Brillinger D.R.: The maximum likelihood approach to the identification of neuronal firing systems. Ann. Biomed. Eng. 16, 3–16 (1988)

    Article  MathSciNet  Google Scholar 

  10. Brillinger D.R.: Remarks concerning graphical models for time series and point processes. Revista de Econometria 16, 1–23 (1996)

    MathSciNet  Google Scholar 

  11. Carrasco M., Chen X.: Mixing and moment properties of various GARCH and stochastic volatitlity models. Econ. Theory 18, 17–39 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chamberlain G.: The general equivalence of Granger and Sims causality. Econometrica 50, 569–581 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cox D.R., Wermuth N.: Multivariate Dependencies—Models, Analysis and Interpretation. Chapman & Hall, London (1996)

    MATH  Google Scholar 

  14. Dahlhaus R.: Graphical interaction models for multivariate time series. Metrika 51, 157–172 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dahlhaus R., Eichler M.: Causality and graphical models in time series analysis. In: Green, P., Hjort, N., Richardson, S. (eds) Highly structured stochastic systems, pp. 115–137. University Press, Oxford (2003)

    Google Scholar 

  16. Dawid A.P.: Conditional independence in statistical theory (with discussion). J. R. Stat. Soc. Ser. B 41, 1–31 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Dawid A.P.: Conditional independence for statistical operations. Ann. Stat. 8, 598–617 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Didelez V.: Graphical models for composable finite markov processes. Scand. J. Stat. 34, 169–185 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Doukhan, P.: Mixing Properties and Examples. In: Lecture Notes in Statistics, vol. 85. Springer, New York (1994)

  20. Drton M., Richardson T.S.: Binary models for marginal independence. J. R. Stat. Soc. Ser. B 70(2), 287–309 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Edwards D.: Introduction to Graphical Modelling, 2nd edn. Springer, New York (2000)

    Book  MATH  Google Scholar 

  22. Eichler M.: A graphical approach for evaluating effective connectivity in neural systems. Philos. Trans. R. Soc. B 360, 953–967 (2005)

    Article  Google Scholar 

  23. Eichler, M.: Graphical modelling of dynamic relationships in multivariate time series In: Winterhalder, M., Schelter, B., Timmer, J. (eds.) Handbook of Time Series Analysis, pp. 335–372. Wiley-VCH, London (2006)

  24. Eichler M.: Granger causality and path diagrams for multivariate time series. J. Econ. 137, 334–353 (2007)

    MathSciNet  Google Scholar 

  25. Eichler, M.: Causal inference from multivariate time series: What can be learned from granger causality. In: Glymour, C., Wang, W., Westerståhl, D. (eds.) Proceedings from the 13th International Congress of Logic, Methodology and Philosophy of Science, College Publications, London (2009)

  26. Eichler, M.: Graphical Gaussian modelling of multivariate time series with latent variables. In: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics. Journal of Machine Learning Research W&CP, vol. 9 (2010)

  27. Engle R.F., Kroner K.F.: Multivariate simultaneous GARCH. Econ. Theory 11, 122–150 (1995)

    Article  MathSciNet  Google Scholar 

  28. Fan J., Yao Q.: Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003)

    Book  MATH  Google Scholar 

  29. Florens J.P., Mouchart M.: A note on noncausality. Econometrica 50, 583–591 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Florens J.P., Mouchart M., Rolin J.M.: Elements of Bayesian Statistics. Marcel Dekker, New York (1990)

    MATH  Google Scholar 

  31. Frydenberg M.: The chain graph Markov property. Scand. J. Stat. 17, 333–353 (1990)

    MathSciNet  MATH  Google Scholar 

  32. Gouriéroux C.: ARCH Models and Financial Applications. Springer, New York (1997)

    Book  MATH  Google Scholar 

  33. Gouriéroux C., Monfort A.: Qualitative threshold ARCH models. J. Econ. 52, 159–199 (1992)

    MATH  Google Scholar 

  34. Granger C.W.J.: Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438 (1969)

    Article  Google Scholar 

  35. Hsiao C.: Autoregressive modeling and causal ordering of econometric variables. J. Econ. Dyn. Control 4, 243–259 (1982)

    Article  Google Scholar 

  36. Koster J.T.A.: On the validity of the Markov interpretation of path diagrams of Gaussian structural equations systems with correlated errors. Scand. J. Stat. 26, 413–431 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Koster J.T.A.: Marginalizing and conditioning in graphical models. Bernoulli 8, 817–840 (2002)

    MathSciNet  MATH  Google Scholar 

  38. Kraft, D.F., Engle, R.F.: Autoregressive conditional heteroscedasticity in multiple time series, unpublished manuscript, Depart. of Economics, UCSD (1982)

  39. Lauritzen S.L.: Graphical Models. Oxford University Press, Oxford (1996)

    Google Scholar 

  40. Levitz M., Perlman M.D., Madigan D.: Separation and completeness properties for AMP chain graph Markov models. Ann. Stat. 29, 1751–1784 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liebscher E.: Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes. J. Time Ser. Anal. 26, 669–689 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lu Z., Jiang Z.: l 1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Stat. Probab. Lett. 51, 121–130 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lynggaard, H., Walther, K.H.: Dynamic modelling with mixed graphical association models. Master’s thesis, Aalborg University (1993)

  44. Moneta, A., Spirtes, P.: Graph-based search procedure for vector autoregressive models. LEM Working Paper 2005/14, Sant’Anna School of Advanced Studies, Pisa (2005)

  45. Oxley L., Reale M., Tunnicliffe Wilson G.: Finding directed acyclic graphs for vector autoregressions. In: Antoch, J. (eds) Proceedings in Computational Statistics 2004, pp. 1621–1628. Physica-Verlag, Heidelberg (2004)

    Google Scholar 

  46. Pearl J.: Probabilistic Inference in Intelligent Systems. Morgan Kaufmann, San Mateo (1988)

    Google Scholar 

  47. Pearl J., Paz A.: Graphoids: a graph based logic for reasoning about relevancy relations. In: Boulay, B.D., Hogg, D., Steel, L. (eds) Advances of Artificial Intelligence—II, pp. 357–363. North-Holland, Amsterdam (1987)

    Google Scholar 

  48. Reale M., Tunnicliffe Wilson G.: Identification of vector AR models with recursive structural errors using conditional independence graphs. Stat. Methods Appl. 10, 49–65 (2001)

    Article  MATH  Google Scholar 

  49. Rothman P: Nonlinear Time Series Analysis of Economic and Financial Data. Kluwer Academic Publishers, Dordrecht (1999)

    Book  MATH  Google Scholar 

  50. Stanghellini, E., Whittaker, J.: Analysis of multivariate time series via a hidden graphical model. In: Proceedings of the 7th International Workshop on Artificial Intelligence and Statistics, pp. 250–254. Morgan Kaufmann, San Mateo, CA (1999)

  51. Talih M., Hengartner N.: Structural learning with time-varying components: tracking the cross-section of financial time series. J. R. Stat. Soc. Ser. B 67, 321–341 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tong H.: Non-Linear Time Series: A Dynamical System Approach. Oxford University Press, Oxford (1993)

    Google Scholar 

  53. Whittaker J.: Graphical Models in Applied Multivariate Statistics. Wiley, Chichester (1990)

    MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank two anonymous referees for their comments and suggestions, which greatly improved the paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD, Maastricht, The Netherlands

    Michael Eichler

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  1. Michael Eichler
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Correspondence to Michael Eichler.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Cite this article

Eichler, M. Graphical modelling of multivariate time series. Probab. Theory Relat. Fields 153, 233–268 (2012). https://doi.org/10.1007/s00440-011-0345-8

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  • Received: 11 May 2006

  • Revised: 14 January 2011

  • Published: 15 February 2011

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-011-0345-8

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Keywords

  • Graphical models
  • Multivariate time series
  • Granger causality
  • Global Markov property

Mathematics Subject Classification (2000)

  • 60G10
  • 62M10
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