Dealing with excess kurtosis via GLM in R I am working in R with a dataset that produces an error distribution under OLS with minimal skew, but excess kurtosis around 2.  After researching the problem on Google for a day I've sorted out that the best approach to performing a regression is to use the Generalized Linear Model with a suitable error model, but I'm stymied trying to figure out which error model to use.  Hypothetically a t-distribution with 7 degrees of freedom would get me the right kurtosis, but given my N is over 1000, I’m skeptical.  Any suggestions on either which error model to use would be appreciated.
 A: To briefly expand on my question:

Can you explain why a large sample size would make you skeptical of something like a $t_7$
  distribution of errors?

It doesn't matter whether you have a large sample or a small sample, the shape of the distribution of the population from which the sample was drawn would be the same.
Large sample size would impact the distribution of a mean of values, not the shape of the distribution of the individual errors. A $t_7$ has a particular form of heavy tail. 
[I would however, be somewhat suspicious of assuming student scores have a t-distribution at all; they're typically discrete and bounded. One might ignore the discreteness in some cases, but the boundedness would tend to be more of an issue.]
A: From your question, I think you may be labouring under a common misunderstanding about the relevance of large N in a statistical model.  The relevance of having a large N is that it allows you to approximate using asymptotic theorems such as the central limit theorem (CLT).  Roughly speaking, this theorem (actually it is a group of theorems) says that when you take a bunch of random variables from any non-heavy-tailed* distribution, and you make a mean quantity out of them, that mean quantity will have a normal distribution when N approaches infinity.  Remember: this theorem asserts normality of the distribution of the mean quantity - the underlying distribution of the individual random variables is still whatever it was.
In the context of regression, this means that the error terms can be non-normal, but even in this case, the coefficient estimators (which are mean-type quantities based on the error terms) for the parameters of your model will be approximately normally distributed when N is large.  If the underlying error distribution is indeed a T(7) distribution, then making N large does not change this.  The CLT does not say that the distribution of underlying error terms becomes normal if N is large.
Another way to think about this is as a reductio ad absurdum.  If you take the view that having large N means that the error distribution should be normal, that is tantamount to saying that we can never observe large numbers of non-normal random variables.
In any case, this means that there is no cause for scepticism of the proposed error distribution based on the size of N.  If the residuals fit closely to a T(7) distribution, then this is a plausible fit.  Not only is there no cause for scepticism, but the fact that you have a large N and you are finding that a T(7) distribution fits this well, means that your empirical estimate of the error distribution is likely to be quite close to the “true” distribution, and so you can be pretty confident that it is a good estimate.
My suggestion would be to fit a regression model with a T-distribution as the error distribution, allowing the DF parameter to be a free parameter that is estimated from the data.  Once you have fitted this model you will get parameter estimates, including an estimate of the DF.  You should then generate an empirical residual density plot (i.e., a plot of the kernel density of the studentised residuals) and superimpose the estimated T distribution over the top of this to see if it looks like a plausible distribution.  You can also try a QQ plot comparing the empirical quantiles of the studentised residuals to the quantiles of the T distribution.  These diagnostic plots will help you see whether the T distribution is a plausible error distribution or not.
Cheers,
Ben.


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*Technical Note: The CLT generally requires that the underlying distribution have a finite variance, which rules out certain heavy-tailed distributions.  In your case, where you have a T-distribution with 7 DF, this distribution has finite variance and so the CLT would apply.

*Another Technical Note: The CLT also requires either that the random variables are independent, or at least not too dependant.  This condition is not relevant here.
