What does T statistics of information coefficient indicate? Hi I am looking for a clear explanation of T statistics concept. Especially in quantitative equity portfolio management context, what does T statistics of information coefficient indicate? How can I tell if it is significant, and if it is, any useful information I could derive from that?
 A: The information coefficient actually is a t-statistic up to scaling. Letting $\hat{\mu}$ be the estimated mean return of the portfolio and $\hat{\sigma}^2$ be the estimated variance, then the Information Coefficient is defined as
$$
\mbox{IC} = \frac{\hat{\mu} - r_{RF}}{\hat{\sigma}},
$$
where $r_{RF}$ is the 'risk free rate' or whatever benchmark you are using. This is, up to scaling, a 1-sample t-test for the null hypothesis $H_0: \mu = r_{RF}$ (if the risk-free rate is varying, or you are using a benchmark, the null is $H_0: \mu - r_{FR}(t) = 0$). The statistic for the 1-sample t-test (see Wikipedia) is
$$
t = \sqrt{n}\frac{\hat{\mu} - r_{RF}}{\hat{\sigma}} = \sqrt{n}\mbox{IC},
$$
where $n$ is the number of samples used to compute $\hat{\mu}$ and $\hat{\sigma}$.
The upshot is that if you multiply $\sqrt{n}$ by the Information Coefficient, you get a t-statistic. Under the null, it is distributed as a central t; under the alternative, it is distributed as a non-central t statistic, where the noncentrality parameter is $\delta = \sqrt{n}(\mu - r_{RF}) / \sigma$, which is root n times the population analogue of the IC. 
There are a number of interesting facts that follow from this:


*

*You can use the non-central t-distribution to find confidence intervals on $\delta$.

*The Information Coefficient statistic is biased, but asymptotically unbiased. For $n < 10$, the bias can be as large as 8 percent.

*The standard error found by Lo, which is essentially $\sqrt{(1 + IC^2/2)/n}$, was first published for the non-central t-distribution by Johnson & Welch in 1940, 60 years prior!

*Corrections for autocorrelation that apply to the t-statistic apply to the IC.


While I have any kind of platform, I would like to note that the Sharpe ratio as devised by Sharpe in 1966 is actually identical (up to a factor of $n / (n-1)$) to the original statistic proposed by Gosset ('Student') in 1906. It was Fisher who added the $\sqrt{n}$ term to make the modern version of the t-statistic. For this reason, I propose the terminology 'Gosset Ratio' instead of 'Sharpe Ratio'.
