Student's t-Distribution

Student's t-distribution is a continuous probability distribution that is symmetric and bell-shaped like the normal distribution but has thicker tails, meaning that values far from the centre are more likely. Its shape is controlled by a single parameter, the degrees of freedom. With few degrees of freedom the tails are very heavy; as the number grows, the distribution approaches the normal curve.

The distribution has two main roles. In statistics it is the basis for testing hypotheses and building confidence intervals about an average when the true variability of the data is unknown and must be estimated from the sample. In finance it is also used directly as a model for asset returns, which show more extreme moves than the normal distribution allows.

The distribution was published in 1908 by William Sealy Gosset, a chemist and statistician at the Guinness brewery in Dublin, who needed methods for drawing conclusions from small batches of experimental data. His employer did not allow staff to publish under their own names, so the paper appeared under the pseudonym Student. Ronald Fisher later placed the result on a rigorous footing and popularized its use.

If $Z$ is a standard normal variable and $V$ is an independent chi-squared variable with $\nu$ degrees of freedom, the ratio below follows a t-distribution with $\nu$ degrees of freedom:

The practical form arises when a sample mean is compared with a hypothesized value:

• $\bar{x}$ is the sample mean.
• $\mu_0$ is the value of the mean under the hypothesis being tested.
• $s$ is the sample standard deviation.
• $n$ is the number of observations, and the degrees of freedom are $n - 1$.

If the data are normally distributed, this t-statistic follows a t-distribution exactly. Using $s$ in place of the unknown true standard deviation adds uncertainty, and the heavier tails account for it. The distribution has a mean of zero and, for $\nu \gt 2$, a variance of $\nu / (\nu - 2)$, which is greater than 1.

A strategy has 12 monthly returns with a mean of 1.2 percent and a sample standard deviation of 2.4 percent. The question is whether the true mean return could be zero. The standard error is $2.4 / \sqrt{12} = 0.693$, so $t = 1.2 / 0.693 = 1.73$.

With 11 degrees of freedom, the critical value for a two-sided test at the 5 percent level is 2.201. Since 1.73 is smaller, the null hypothesis of a zero mean is not rejected. Had the normal distribution been used, the threshold would have been 1.96, which is easier to pass. The gap between 2.201 and 1.96 is the allowance the t-distribution makes for the small sample. With 30 degrees of freedom the critical value is 2.042, and with several hundred it is practically 1.96.

The t-distribution underlies the t-test for one mean, for the difference between two means and for paired observations. It gives the confidence interval for a mean, and it is the reference distribution for testing whether an individual coefficient in a linear regression differs from zero. The p-value reported beside each regression coefficient by statistical software is computed from it.

In trading research these tests are the usual way to judge whether an average return, an estimated alpha or a factor exposure is distinguishable from zero. For large samples the z-test, based on the normal distribution, gives nearly the same answer.

Daily returns on shares, currencies and commodities have fat tails: moves of four or five standard deviations occur far more often than the normal curve predicts. A t-distribution with a small number of degrees of freedom, commonly between three and six for daily data, reproduces this behaviour while remaining simple to work with. The excess kurtosis of the distribution is $6 / (\nu - 4)$ for $\nu \gt 4$, so five degrees of freedom imply an excess kurtosis of 6, against zero for the normal distribution.

Risk managers therefore use it to compute value at risk and related tail measures, obtaining larger loss estimates at high confidence levels than a normal model would give. Volatility models such as GARCH are often fitted with t-distributed shocks, and a multivariate version is used to model the tendency of assets to suffer extreme losses together.

The exact result for the t-statistic assumes that the observations are independent and normally distributed. Financial returns violate both conditions to some degree, so reported significance levels are approximate, and adjustments for serial correlation and changing volatility are common. As a model of returns, the t-distribution is symmetric and so cannot capture skewness; skewed variants exist for that purpose. With very few degrees of freedom some moments do not exist: the variance is infinite at two or fewer, which makes standard risk measures unusable.