Granger causality on horizontal sum of Boolean algebras

TL;DR

This paper introduces a novel mathematical model of causality using the horizontal sum of Boolean algebras, modifying the traditional Kolmogorovian probability framework to better analyze causal relationships.

Contribution

It proposes a new causality model based on the horizontal sum of Boolean algebras, offering an alternative to the standard product-based approach.

Findings

Developed a modified Kolmogorovian probability model

Applied the model to causality analysis

Provided theoretical insights into Boolean algebra structures

Abstract

The intention of this paper is to discuss the mathematical model of causality introduced by C.W.J. Granger in 1969. The Granger's model of causality has become well-known and often used in various econometric models describing causal systems, e.g., between commodity prices and exchange rates. Our paper presents a new mathematical model of causality between two measured objects. We have slightly modified the well-known Kolmogorovian probability model. In particular, we use the horizontal sum of set $\sigma$-algebras instead of their direct product.