Linear regression with fat-tailed errors I'm testing a linear model that explains stock returns with some contemporaneous factors; the model is assumed to satisfy OLS assumptions except that the errors (i.e., unexplained stock returns) have fat tails: let's say they are described by an alpha stable or Student's t distribution.
How should I estimate the model? I assume I should transform the data, or use a generalized linear model? I'm willing to be liberal with the distribution assumptions, as long as the calibration of the distribution and the estimation itself are not too complex.
 A: What software are you using? If you use Stan, for example, one of the examples in the manual (Section 11.3) uses a t-distribution instead of normal. You can use Stan from within R, with the package rstan -- Stan comes bundled with rstan, so no need to download and build Stan separately.
As Peter Flom says, that's a type of robust regression, so perhaps a straight-up robust regression (most any statistical package will have one or two versions) would work.
A: If the problem is only the fat tail, (i.e. the residuals are still constant mean equal to 0, constant variance and symmetric etc.), the OLS coefficients are still BLUE.
So this method is still the "Best".
Intuitively, I think it is still equal to MLE.
The problem resuls that the t-test and F-test are not valid, and the AIC in the result is not the true AIC value.
For such tests, maybe you can calculate the likelihood function (with t family or others) and employ a deviance test.
Or more simply, compare the cross-validation errors with/without that variable.
