# What's a good prior distribution for degrees of freedom in a t distribution?

I want to use a t distribution to model short interval asset returns in a bayesian model. I'd like to estimate both the degrees of freedom (along with other parameters in my model) for the distribution. I know that asset returns are quite non-normal, but I don't know too much beyond that.

What is an appropriate, mildly informative prior distribution for the degrees of freedom in such a model?

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A t distribution might not be a good choice, because it is symmetric whereas asset returns tend to have strong skew. At a minimum, consider modeling the logarithms of the returns rather than the returns themselves. – whuber Jan 24 '11 at 19:28
Yeah, that's a good point, I was thinking about that in the back of my mind, but this question is still of interest to me. – John Salvatier Jan 24 '11 at 19:34
Do you have a truly huge amount of data? I think it's more common even in Bayesian modelling to fix the df and try different values as a sensitivity analysis. – onestop Jan 24 '11 at 20:11
I do have a pretty large quantity of data, but it may be that this is the best approach. Submit as an answer and I'll vote you up and accept if no one provides a better solution. – John Salvatier Jan 24 '11 at 21:07
I would try using the Laplace distribution for asset returns, also called the "double exponential" is stats-world, and "variance-gamma" in Finance world. – probabilityislogic Mar 27 '11 at 9:54

On page 372 of ARM, Gelman and Hill mention using a uniform distribution on the inverse of DF between 1/DF = .5 and 1/DF = 0.

Specifically, in BUGS, they use:

nu.y <- 1/nu.inv.y
nu.inv.y ~ dunif(0,.5)

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Here's an article that might help.

http://www.portfolioprobe.com/2011/01/12/the-number-1-novice-quant-mistake/

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 note that using t-distribution for GARCH returns is quite popular in financial econometrics. So the question is pretty valid and does not imply that the link applies to the poster. – mpiktas Jan 25 '11 at 7:09 @mpiktas I am not sure, but bill_080's point might have been that at the very beginning of the article they use n = 6 for their t-distribution. It can be useful to know what people consider reasonable values. – John Salvatier Jan 25 '11 at 17:57 @John, you are correct, n=6 was my point. I tried several schemes that you might use in fitting n for your data and a t distribution (using fitdist from package fitdistrplus), but I couldn't configure a way that I thought was worth posting. – bill_080 Jan 25 '11 at 19:47