Ljiljana Petrovic



Generalization of Granger Causality in Continuous Time

pdf PDF


The paper considers a statistical concepts of causality in continuous time between flows of information and between stochastic processes which is based on Granger’s definitions of causality. More precisely, we will see how conditional orthogonality and conditional independence can serve as a basis for a general probabilistic theory of causality for both stochastic processes and single events. These results are motivated by causality relationship between filtrations ”(Gt) is a cause of (Et) within (Ft)” and which is based on Granger’s definition of causality. Also, we consider causality relationships between s-fields (filtrations) associated by stopping times, which are applicable to the stopped processes (see Petrovic et al. 2016). Then we give some basic properties of causality up to some stopping time.


Hilbert space, filtration, causality, stopping time, stopped process


[1] H. Akaike, Stochastic Theory of Minimal Realization, IEEE Trans. Autom. Control 19, 6, 1974, 667–674. [1] H. Akaike, Stochastic Theory of Minimal Realization, IEEE Trans. Autom. Control 19, 6, 1974, 667–674. 

[2] O. Aalen, A. Frigressi, What can statistics contribute to a causal understanding? Scand. J. Statist. 34, 2007, 155–168 

[3] R.R. Bahadur, Sufficiency and Statistical Decision Functions, Annals of Mathematical Statistics, Vol. 25, 1954, 423–462. 

[4] P. Bremaud and M. Yor, Changes of Filtrations and of Probability measures, Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 1978, 269– 295. 

[5] D. Commenges, A. Gegout-Petit, A general dy- ´ namical statistical model with causal interpretation, J. R. Statist. Soc. B 71, 2009, 719–736 

[6] F. Comte, E. Renault, Noncausality in Continuous Time Models, Econometric Theory, Vol. 12, 1996, 215–256 

[7] R. J. Elliot, Stochastic Calculus and Applications (Springer-Verlag, New York, 1982). 

[8] J.P. Florens, D. Fougeres, Noncausality in Con- ` tinuous Time, Econometrica 64 1996, 1195– 1212. 

[9] J. P. Florens, M. Mouchart, J.M. Rolin, Elements of Bayesian Statistics (Marcel Dekker, New York, 1990). 

[10] J. Geweke, R.C. Marshall, G.A. Zarkin, ”Mobility Indices in Continuous Time Markov Chains”, Econometrica, Vol.54, 1986, 1407–1423. 

[11] J.B. Gill, Lj. Petrovic, Causality and Stochas- ´ tic Dynamic Systems, SIAM J. Appl. Math. 47, 1987, 1361–1366 . 

[12] C.W.J. Granger, Investigation Causal Relations by Econometric Models and Cross Spectral Methods, Econometrica 37, 1969, 424–438. 

[13] C. W. J., Granger, P. Newbold, Forecasting Economic Time Series (Academic Press, New York, 1977). 

[14] W.J. Hall, R.A. Wijsman, J.K. Gosh, The Relationship between Sufficiency and Invariance with Applications in Sequential Analysis, Annals of Mathematical Statistics, Vol. 36, 1965, 575–614. 

[15] J.P. Heckman, B. Singer, ”Economic Duration Analysis”, Journal of Econometrics, 24, 1982, 563–132. 

[16] D.N. Hoover, J.H. Keisler, Adapted Probability Distributions, Trans. Am.Math.Soc., 286, 1984, 159–201 

[17] J. Jacod, A. N. Shiryaev, 2002. Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin. 

[18] J. R. McCrorie, M. J. Chambers, Granger causality and sampling of economic processes, Journal of Econometrics 132, 2006, 311–336. 

[19] M. P. Kean, Brownian Motion with Severaldimensional Time, Theory Probab. Appl., VIII, 4, 1963, 203-210. 

[20] O. Knill, Probability Theory and Stochastic Processes with Applications, Overseas Press, 2009. 

[21] A. Lindquist, A., G. Picci, Realization Theory for Multivariate Stationary Gaussian Processes, SIAM Control and Optimization 20, 6, 1985, 809–857. 

[22] A. Melino, ”Estimation of Continuous-Time Models in Finance”, Advances in Econometrics, Sixth World Congress, Econometricic Society Monographs, ed. by C.A. Sims, Cambridge; Cambridge University Press, 1994, 313–354. 

[23] R.C. Merton, (1990), ”Continuous- Time Finance”, Oxford: Basil Blackwell. 

[24] P.A. Mykland, Statistical Causality, Statistical report No. 14, University of Bergen, 1986. 

[25] Lj. Petrovic, Causality and Stochastic Realiza- ´ tion Problem, Publ. Inst. Math. 45(59), 1989, 203–210. 

[26] Lj. Petrovic, Causality and Markovian Repre- ´ sentations, Statist. Probab. Lett. 29, 1996, 223– 227. 

[27] Lj. Petrovic, Causality and Markovian Reduc- ´ tions and Extensions of a Family of Hilbert Spaces, J. Math. Systems, Estimat. Control 8, 1998, 12 pp. 

[28] Lj. Petrovic, Statistical Causality and Stochas- ´ tic Dynamic Systems, International Journal of Applied Mathematics and Informatics, (ISSN 2074–1278), Issue 3, Volume 5, 153–156, 2011. 

[29] Lj. Petrovic, S. Dimitrijevi ´ c, Invariance of sta- ´ tistical causality under convergence, Statist. Probab. Lett. 81, 2011, 1445–1448. 

[30] Lj. Petrovic, S. Dimitrijevi ´ c, Statistical causality ´ and adapted distribution, Czhechoslovak Mathematica Journal, ISSN 0011-4642 Springer Verlag, Vol. 61, No. 3, 2011, 827–843. 

[31] Lj. Petrovic, S. Dimitrijevi ´ c, Causality with fi- ´ nite horizon of the past in continuous time, Statist. Probab. Lett. 82, 2012, 1219–1223. 

[32] Lj. Petrovic, D. Stanojevi ´ c, Statistical Causali- ´ ty, Extremal Measures and Weak Solutions of Stochastical Differential Equations With Driving Semimartingales J. Math. Model. Algor. 9, 2010, 113–128. 

[33] Lj. Petrovic, D.Valjarevi ´ c, Statistical Causality ´ and stable subspaces of Hp , Bull. Aust. Math. Soc., (2012), 1–9. 

[34] Lj. Petrovic, S. Dimitrijevi ´ c, D. Valjarevi ´ c,´ Granger Causality and Stopping Times, Lithuanian Mathematical Journal, 56(3), 410-416, 2016. 

[35] C. van Putten, J.H. van Schuppen, On Stochastic Dynamic Systems, International Symposium on Mathematical Theory of Networks and Systems, Vol. 3 (Delft 1979), Western Periodical, North Hollywood, Calif., 1979, 350-356. 

[36] Rozanov, Yu.A. (1977), On Markovian Extensions of Random Process, Theory Probab. Appl., 22, 1, 194–199. 

[37] Rozanov, Yu.A. (1982), Markov Random Fields, Springer-Verlag, Berlin, New York, Heidelberg. 

[38] D. Valjarevic, Lj. Petrovi ´ c, Statistical causality ´ and orthogonality of local martingales, Statist. Probab. Lett. 82, 2012, 1326–1330.

Cite this paper

Ljiljana Petrovic. (2016) Generalization of Granger Causality in Continuous Time. Mathematical and Computational Methods, 1, 281-286


Copyright © 2016 Author(s) retain the copyright of this article.
This article is published under the terms of the Creative Commons Attribution License 4.0