Interpreting Shannon Entropy in Rolling Stock Return Windows

Summary

The document asks whether Shannon entropy calculated over successive windows of stock returns should rise or fall as time advances. It defines entropy as a measure of uncertainty in a probability distribution and contrasts rolling windows, each estimated from a different segment of the return series, with an expanding window that accumulates observations over time. It also asks whether measured entropy mainly reflects volatility, since both relate to the distribution of returns, and whether thermodynamic ideas about the passage of time apply.

No answer, calculation, dataset, or empirical result is included, so the questions remain open. The distinction between a changing return distribution and the estimation process matters: rolling-window entropy can vary as the underlying distribution changes, while expanding-window estimates also depend on the observations accumulated and the estimator used. The document raises a useful statistical question, but does not establish a time trend or a direct relationship between financial entropy and thermodynamic entropy.

Key ideas

• Shannon entropy measures uncertainty in a probability distribution and can be estimated from return observations.
• Rolling windows produce separate entropy estimates for successive segments of a return series.
• The document asks whether changes in entropy reflect changing volatility or other distributional features.
• Expanding-window estimates raise a distinct question because each estimate includes accumulated history.
• No evidence is provided for a consistent time trend or a link to thermodynamic entropy.

Tags

# Does the Shannon entropy of stock returns change over time?


# Does the Shannon entropy of stock returns change over time?












Shannon entropy, $H(X) = -\sum_{i=1}^n p(x) \ln p(x)$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram.

If we take rolling window segments, or snapshots, of a full-sample time series of stock returns $X$, can we expect the Shannon entropy of these windows to consistently increase or decrease as $t\rightarrow \infty$? i.e. each window has its own pdf.

or would their entropy merely be a function of the volatility (clustering) in each window, given that the spread of a distribution (volatility) and its randomness (entropy) are inextricably linked?

In thermodynamics, time entropy naturally grows with time, so mustn't there be a connection between the Shannon entropy and time entropy of a stock's returns?

How about the entropy of stock returns using expanding windows instead?

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