Excess Kurtosis in the error term.

[Kevin Forbes]

I am modelling the residual errors terms in my time-series model using the Student t distribution. This distribution allows for more kurtosis (‘heavy tailedness’) than the Gaussian distribution. Specifically, the level of kurtosis that is accommodated by this distribution in excess of the Gaussian’s level of three equals 6/(v - 4) provided that v > 4, where v is the number of degrees of freedom (Harvey, 2013, p. 20). In this case, the number of degrees of freedom does not refer to the sample size minus the number of estimated parameters, but instead is a “shape” parameter for the distribution.

In my current model, the value of "v" is 2.87. Any sense of what this means about the level of excess kurtosis?

Here is a reference to Harvey's book

Harvey, A. C. (2013). Dynamic models for volatility and heavy tails: With applications to financial and economic time series. New York: Cambridge University Press. https://doi.org/10.1017/CBO9781139540933

[Maarten Buis]

You can see what it means in a graph:

twoway function t = tden(2.87, x), range(-6 6) || function normal = normalden(x), range(-6 6)

[Nick Cox]

Cross-posted at https://stats.stackexchange.com/ques...the-error-term

[Jeff Wooldridge]

If you rely on that estimate of v then the situation is not promising. It means the error distribution does not have a finite second moment, which means the asymptotic properties of the MLE are not known — unless there’s recent work. I suppose you could simulate to see.

since the distribution doesn’t have a finite fourth moment, kurtosis is not defined.

[Maarten Buis]

I seem to remember Yulia Marchenko (StataCorp) worked on estimation with t-distributions. Maybe she can shed some light on this?

[Maarten Buis]

I know now. It was https://www.jstor.org/stable/23239671

I wish I remembered it because of its quality (which I am sure it has aplenty), but it was because I got a picture of my oldest son reading that JASA article to his teddy bear (he held the journal upside down).