Entropy Analysis of Crisis Phenomena for DJIA Index

Abstract

The Dow Jones Industrial Average (DJIA) index for the 125-year-old (since 1896) history has experienced many crises of different nature and, reflecting the dynamics of the world stock market, is an ideal model object for the study of quantitative indicators and precursors of crisis phenomena. In this paper, the classification and periodization of crisis events for the DJIA index have been carried out; crashes and critical events have been highlighted. Based on the modern paradigm of the theory of complexity, a spectrum of entropy indicators and precursors of crisis phenomena have been proposed. The entropy of a complex system is not only a measure of uncertainty (like Shannon's entropy) but also a measure of complexity (like the permutation and Tsallis entropy). The complexity of the system in a crisis changes significantly. This fact can be used as an indicator, and in the case of a proactive change as a precursor of a crisis. Complex systems also have the property of scale invariance, which can be taken into account by calculating the Multiscale entropy. The calculations were carried out within the framework of the sliding window algorithm with the subsequent comparison of the entropy measures of complexity with the dynamics of the DJIA index itself. It is shown that Shannon's entropy is an indicator, and the permutation and Tsallis entropy are the precursors of crisis phenomena to the same extent for both crashes and critical events.