Shannon entropy and high frequency financial time series

The thesis considers the problem of evaluating a degree of market efficiency. In a weak form of the Efficient Market Hypothesis (EMH), it is impossible to predict a future price of an asset based on prices available at the current time. The Shannon entropy is used as a measure of the randomness of price returns. To determine the degree of market efficiency, I propose a method for filtering out data regularities. Data regularities are empirical properties of price returns that make prices more predictable. I investigate price staleness that generates 0-returns as one of the data regularities. I show that inefficient time intervals have a larger amount of 0-returns than efficient time intervals. Then, I propose a method for filtering out spurious 0-returns.First, I investigate Exchange Traded Funds (ETFs) traded at the New York Stock Exchange. With a significance level of 1% for testing EMH , the degree of inefficiency of the ETF market for weekly time intervals at a one-minute frequency is equal to 1.35%. The degree of market inefficiency calculated for monthly intervals is about 11%. I find statistically significant co-inefficiency for all considered lengths of time intervals: weeks, months, quarters.Second, I investigate the efficiency of the Moscow stock exchange. The degree of inefficiency for the Moscow Stock Exchange is 82%. I show that months where the randomness of the stock prices attains its minimum group together. I determine what behavior of prices repeats most often for inefficient time intervals. With the Kullback-Leibler distance, I cluster stocks into three groups. For instance, I show that banks and gas companies cluster together. I introduce the discretization describing co-movement of prices. Estimating the entropy of the obtained symbolic sequences, I point out that market inefficiency displays some dependence from the sector to which companies belong.Third, I propose a hypothesis testing procedure to test the null hypothesis of equality of entropies. I find the optimal length of the rolling window used for estimating the time-varying Shannon entropy by optimizing a novel self-consistent criterion. I use the novel methodology to test for time-varying regimes of entropy. I empirically show the existence of periods of market inefficiency for meme stocks.Finally, I consider the ultra-high frequency of price returns. I find theoretical quantiles of gamma distribution that help to quickly test for randomness of the data.