Variance of entropy for testing time-varying regimes with an application to meme stocks

TL;DR

This paper develops a hypothesis testing method to detect significant changes in the Shannon entropy of time series, with applications to identifying market inefficiencies in meme stocks during 2020-2021.

Contribution

It introduces an unbiased variance approximation for Shannon entropy estimators and a criterion for optimal window length in time-varying entropy analysis.

Findings

Detected periods of market inefficiency in meme stocks.

Identified significant entropy drops during sharp price increases.

Validated the methodology on real stock data.

Abstract

Shannon entropy is the most common metric to measure the degree of randomness of time series in many fields, ranging from physics and finance to medicine and biology. Real-world systems may be in general non stationary, with an entropy value that is not constant in time. The goal of this paper is to propose a hypothesis testing procedure to test the null hypothesis of constant Shannon entropy for time series, against the alternative of a significant variation of the entropy between two subsequent periods. To this end, we find an unbiased approximation of the variance of the Shannon entropy's estimator, up to the order O(n^(-4)) with n the sample size. In order to characterize the variance of the estimator, we first obtain the explicit formulas of the central moments for both the binomial and the multinomial distributions, which describe the distribution of the Shannon entropy. Second, we find the optimal length of the rolling window used for estimating the time-varying Shannon entropy by optimizing a novel self-consistent criterion based on the counting of significant variations of entropy within a time window. We corroborate our findings by using the novel methodology to test for time-varying regimes of entropy for stock price dynamics, in particular considering the case of meme stocks in 2020 and 2021. We empirically show the existence of periods of market inefficiency for meme stocks. In particular, sharp increases of prices and trading volumes correspond to statistically significant drops of Shannon entropy.