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

Abstract

Shannon entropy is the most common metric for assessing the degree of randomness of time series in many fields, ranging from physics and finance to medicine and biology. Real-world systems are typically non-stationary, leading to entropy values fluctuating over time. This paper proposes a hypothesis testing procedure to test the null hypothesis of constant Shannon entropy in time series data. The alternative hypothesis is a significant variation in entropy between successive periods. To this end, we derive an unbiased sample entropy variance, accurate up to the order \(O(n^{-4})\) with n the sample size. To characterize the variance of the sample entropy, we first provide explicit formulas for the central moments of both binomial and multinomial distributions describing the distribution of the sample entropy. Second, we identify the optimal rolling window length to estimate time-varying Shannon entropy. We optimize this choice using a novel self-consistent criterion based on counting significant entropy variations over time. We corroborate our findings using the novel methodology to assess time-varying regimes of entropy for stock price dynamics by presenting a comparative analysis between meme and IT stocks in 2020 and 2021. We show that low entropy values correspond to periods when profitable trading strategies can be devised starting from the symbolic dynamics used for entropy computation, namely periods of market inefficiency.