Measuring market efficiency: The Shannon entropy of high-frequency financial time series

When prices reflect all available information, the price dynamics is a martingale and the market is said to be efficient. However, much empirical evidence supports the conclusion about the inefficiency of financial markets, especially at high-frequency timescales. We investigate the sources and dynamics of the inefficiency of the ETF market at a 1 min timescale by proposing a computational methodology for a genuine estimation of the Shannon entropy. Since several sources of regularity lead to the detection of apparent inefficiencies, we build a multi-step filtering method, which allows (i) to remove the seasonality of volatility and heteroskedasticity, (ii) to detect and remove spurious effects due to price staleness, and (iii) to filter out microstructure noise. We corroborate our findings with an extensive analysis of the ETF market. We conclude that, after removing all known patterns of regularity, the market is not efficient at a one-minute time scale and on a weekly basis; however, the signal is weak.