  We use the statistical properties of Shannon entropy estimator and
Kullback-Leibler divergence to study the predictability of ultra-high frequency
financial data. We develop a statistical test for the predictability of a
sequence based on empirical frequencies. We show that the degree of randomness
grows with the increase of aggregation level in transaction time. We also find
that predictable days are usually characterized by high trading activity, i.e.,
days with unusually high trading volumes and the number of price changes. We
find a group of stocks for which predictability is caused by a frequent change
of price direction. We study stylized facts that cause price predictability
such as persistence of order signs, autocorrelation of returns, and volatility
clustering. We perform multiple testing for sub-intervals of days to identify
whether there is predictability at a specific time period during the day.
