This paper first surveys empirical patterns that appear in asset returns. The main patterns are the following.
Returns are hard to predict out of sample.
The correlation dimension is rather large and is unstable. The cross autocorrelation function of detrended volume and volatility falls off rapidly in leads and lags with a large peak at 0. More sophisticated bookstrapping type tests have found evidence of some conditional predictability of returns out of sample provided the conditioning set is chosen carefully. But rational expectations beliefs are costly and compete with other types of beliefs in generating net trading profits as the system moves forward in time. The paper briefly reviews some studies of equilibrium returns, volume of traditing, and volatility of returns that is consistent with the stylized facts reviewed above.