Examples of Student's T-distribution in real world empirical data? I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points in the tails. On the other hand, the t-distribution (df=2) fits virtually perfectly.
I need to know how to explain it if I wish to use it for modelling purposes.
Is there a logic to this, or am I "over-fitting" the data? I would really like to know why this is happening and where this can occur in other real-world samples.
Note: I'm sorry I cannot share the exact nature of the data, but I can say it has over 15,000 data points.
 A: Actually it is more likely that the original data would be fit to a time series.  If the model is some function of past values of the series + a random normal error and you have fit the corect form to the model then the sample residuals will be approximately normal but not exactly because of the parameter estimates being used in place of the unknown parameters.  Why in you particular example these residuals look like they are samples from a t distribution is probably just coincidental.  If the model is correct the residuals would be only slightly correlated and would have approximately a 0 mean and constant variance.  It is more reasonable to think that the model is at least slightly off and so there is some structure to the residuals.
A: I think I figured out the reason, but please correct me if I'm wrong.
Since it's a forecast error, the original forecast will likely be based on a regression. Since each independent (input) variable will have an error associated with it, the forecast error will be the weighed sum of the errors.
What this implies, is that the distribution is probably not a t-distribution, but rather multiple superimposed normal distributions which look like a t-distributions.
A: For a linear forecasting model with unknown variance and normally distributed noise, the forecasting error should indeed theoretically follow the $t$ distribution (not the normal distribution). See, e.g, here.
However, with over 15,000 data points, df=2 is unexpected (usually the degrees of freedom df would be similar to the number of data points), unless you are fitting the forecast model to only a few observations at a time. If the noise has longer tails than a normal distribution , though, a $t$ distribution with few df (i.e., long-tailed) could well give a reasonable fit.
