Noncausality and Marginalization of Markov Processes

In this paper it is shown that a subprocess of a Markov process is markovian if a suitable condition of noncausality is satisfied. Furthermore, a markovian condition is shown to be a natural condition when analyzing the role of the horizon (finite or infinite) in the property of noncausality. We also give further conditions implying that a process is both jointly and marginally markovian only if there is both finite and infinite noncausality and that a process verifies both finite and infinite noncausality only if it is markovian. Counterexamples are also given to illustrate the cases where these further conditions are not satisfied.

[1]  C. Granger.  Testing for causality: a personal viewpoint , 1980 .

[2]  H. Teicher.  Identifiability of Mixtures of Product Measures , 1967 .

[3]  On Statistics Independent of a Sufficient Statistic: Basu's Lemma , 1975 .

[4]  Jean-Pierre Florens,et al.  Elements of Bayesian Statistics , 1990 .

[5]  Jean-Marie Dufour,et al.  Testing Causality Between Two Vectors in Multivariate Arma Models , 1991 .

[6]  C. Sims.  Money, Income, and Causality , 1972 .

[7]  C. Granger.  Investigating Causal Relations by Econometric Models and Cross-Spectral Methods , 1969 .

[8]  J. Geweke,et al.  Measurement of Linear Dependence and Feedback between Multiple Time Series , 1982 .

[9]  Jean-Pierre Florens,et al.  A Linear-theory for Noncausality , 1985 .

[10]  John Geweke,et al.  Comparing alternative tests of causality in temporal systems: Analytic results and experimental evidence☆ , 1983 .

[11]  J. Florens,et al.  A Note on Noncausality , 1982 .

[12]  Larry D. Haugh,et al.  Causality in temporal systems: Characterization and a survey , 1977 .

[13]  Marc Yor,et al.  Changes of filtrations and of probability measures , 1978 .

[14]  La modélisation économétrique des transitions individuelles sur le marché du travail , 1994 .

[15]  C. Gouriéroux,et al.  Kullback Causality Measures , 1987 .

[16]  C. Granger,et al.  Forecasting Economic Time Series. , 1988 .

[17]  Michel Mouchart,et al.  A note on conditional independence with statistical applications , 1985 .

[18]  Quang Vuong,et al.  Tests of Noncausality under Markov Assumptions for Qualitative Panel Data , 1986 .

[19]  Michel Mouchart,et al.  Conditioning in dynamic models , 1985 .

[20]  C. Sims.  MACROECONOMICS AND REALITY , 1977 .

[21]  Michel Mouchart,et al.  A note on conditional independence , 1984 .

[22]  L. Breiman,et al.  Consistent Estimates and Zero-One Sets , 1964 .

[23]  On statistics independent of a sufficient statistic , 1957 .