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Monotone dependence in graphical models for multivariate Markov chains

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  • Roberto Colombi

  • Sabrina Giordano

Abstract

We show that a deeper insight into the relations among marginal processes of a multivariate Markov chain can be gained by testing hypotheses of Granger noncausality, contemporaneous independence and monotone dependence. Granger noncausality and contemporaneous independence conditions are read off a mixed graph, and the dependence of an univariate component of the chain on its parents—according to the graph terminology—is described in terms of stochastic dominance criteria. The examined hypotheses are proven to be equivalent to equality and inequality constraints on some parameters of a multivariate logistic model for the transition probabilities. The introduced hypotheses are tested on real categorical time series. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Roberto Colombi & Sabrina Giordano, 2013. "Monotone dependence in graphical models for multivariate Markov chains," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(7), pages 873-885, October.
    • Handle: RePEc:spr:metrik:v:76:y:2013:i:7:p:873-885
    DOI: 10.1007/s00184-012-0421-9
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    References listed on IDEAS

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    1. Thomas Richardson, 2003. "Markov Properties for Acyclic Directed Mixed Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 145-157, March.
    2. Eichler, Michael, 2007. "Granger causality and path diagrams for multivariate time series," Journal of Econometrics, Elsevier, vol. 137(2), pages 334-353, April.
    3. Colombi, R. & Giordano, S., 2012. "Graphical models for multivariate Markov chains," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 90-103.
    4. Chamberlain, Gary, 1982. "The General Equivalence of Granger and Sims Causality," Econometrica, Econometric Society, vol. 50(3), pages 569-581, May.
    5. Steen A. Andersson & David Madigan & Michael D. Perlman, 2001. "Alternative Markov Properties for Chain Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 33-85, March.
    6. Anna Gottard, 2007. "On the inclusion of bivariate marked point processes in graphical models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(3), pages 269-287, November.
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