A statistic for measuring the influence of side information in investment

Side information can increase the growth rate of wealth in stock market investment. The question is: when is the side information useful and when is it illusory? We propose a statistic to test this. We compare the wealth achieved by the best constant rebalanced portfolio in hindsight on the entire stock market sequence to the wealth resulting from the best constant rebalanced portfolio in hindsight on the subsequences of stocks identified by the states of the side information sequence. If one can't make money from side information given perfect hindsight one never can. We prove, under the null hypothesis of independence of the side information sequence from the stock sequence, that the logarithm of the ratio of optimal wealth with and without side information is asymptotically distributed as half a chi-squared random variable with (d - 1)(m - 1) degrees of freedom, where d is the number of states of side information and m is the number of stocks. This statistic has an asymptotic distribution that is independent of the particular stock sequence or the distribution of the side information sequence. Thus the use of side information can be accepted when the wealth ratio exceeds a certain fixed level given by the appropriate quantile of the chi-squared distribution