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We determine the amount of information contained in a time series of price returns at a given time scale, by using a widespread tool of the information theory, namely the Shannon entropy, applied to a symbolic representation of this time series. By deriving the exact and the asymptotic distribution of this market information indicator in the case where the efficient market hypothesis holds, we develop a statistical test of market efficiency. We apply it to a real dataset of stock indices, single stocks, and cryptocurrencies, for which we are able to determine at each date whether the efficient market hypothesis is to be rejected, with respect to a given confidence level.

A statistical test of market efficiency based on information theory