Causality and stochastic realization

Ljiljana Petrovic

doi:10.1155/ijmms.2005.349

Abstract

We consider a problem (that follows directly from realization problem) on finding Markovian representations for a given family of Hilbert spaces such that each of these two families provides exactly the same amount of information about some other family of Hilbert spaces.

Highlights

The study of Granger causality has been mainly preoccupied with time series
We will instead concentrate on continuous time processes
In the first part of this paper, we give a generalization of a causality relationship “G is a cause of E within H” which was first given in [4] and which is based on Granger’s definition of causality [2]

Summary

Introduction

The study of Granger causality has been mainly preoccupied with time series. We will instead concentrate on continuous time processes. Many systems to which it is natural to apply tests of causality take place in continuous time. This is generally the case within economy. In the first part of this paper, we give a generalization of a causality relationship “G is a cause of E within H” which (in terms of σ-algebras) was first given in [4] and which is based on Granger’s definition of causality [2]. Since our results do not depend on probability distribution, we deal with arbitrary Hilbert spaces instead of those generated by Gaussian processes. It is clear that all the results of this paper can be extended on the σ-algebras generated by finite-dimensional Gaussian random variables.

Preliminary notions and definitions

Causality and stochastic dynamic systems