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.
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References
Andersson S, Madigan D, Perlam M (2001) Alternative Markov properties for chain graphs. Scand J Stat 28:33–85
Bartolucci F, Colombi R, Forcina A (2007) An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints. Stat Sin 17:691–711
Basawa IV, Prakasa Rao BLS (1980) Statistical inference for stochastic processes. Academic Press, New York
Cazzaro M, Colombi R (2009) Multinomial-Poisson models subject to inequality constraints. Stat Model 9(3):215–233
Chamberlain G (1982) The general equivalence of Granger and Sims causality. Econometrica 50(3): 569–582
Colombi R, Giordano S (2012) Graphical models for multivariate Markov chains. J Multivar Anal 107: 90–103
Colombi R, Giordano S, Cazzaro M (2012) R-package hmmm: hierarchical multinomial marginal models. http://CRAN.R-project.org/package=hmmm
Cox DR, Wermuth N (1996) Multivariate dependencies—models, analysis and interpretation. Chapman and Hall, London
Dawid AP (1979) Conditional independence in statistical theory (with discussion). J R Stat Soc Ser B 41:1–13
Didelez V (2006) Graphical models for composable finite Markov processes. Scand J Stat 34:169–185
Douglas R, Fienberg SE, Lee MT, Sampson AR, Whitaker LR (1990) Positive dependence concepts for ordinal contingency tables. Institute of Mathematical Statistics. Lecture Notes-Monograph Series: Topics in Statistical Dependence 16:189–202
Eichler M (2007) Granger causality and path diagrams for multivariate time series. J Econom 137:334–353
Eichler M (2012) Graphical modelling of multivariate time series. Prob Theory Relat Fields 153:233–268
Glonek GJN, Mccullagh P (1995) Multivariate logistic models. J R Stat Soc B 57:533–546
Gottard A (2007) On the inclusion of bivariate marked point processes in graphical models. Metrika 66: 269–287
Lauritzen SL (1996) Graphical models. Clarendon Press, Oxford
Kijima M (1997) Markov chains and stochastic modeling. Chapman-Hall, London
Marchetti GM, Lupparelli M (2011) Chain graph models of multivariate regression type for categorical data. Bernoulli 17:736–753
Richardson T (2003) Markov properties for acyclic directed mixed graphs. Scand J Stat 30:145–157
Shaked M, Shantikumar JG (1994) Stochastic orders and their applications. Academic Press, Boston
Silvapulle MJ, Sen PK (2005) Constrained statistical inference. Wiley, New-Jersey
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Colombi, R., Giordano, S. Monotone dependence in graphical models for multivariate Markov chains. Metrika 76, 873–885 (2013). https://doi.org/10.1007/s00184-012-0421-9
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DOI: https://doi.org/10.1007/s00184-012-0421-9