Excess kurtosis in the error term I am modelling the residual errors terms in my time-series model using Student's $t$-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 $\dfrac{6}{\nu - 4}$ provided that $\nu > 4$, where $\nu$ 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 $\nu$ 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
 A: For the t-distribution you need $\nu>k$ for $E(X^k)$ to exist. So you need $\nu>4$ for a finite fourth moment. Then you can calculate the kurtosis:
\begin{align}
\kappa=\frac{E((X-E(X))^4)}{V(X)^2}=\frac{E(X^4)}{(E(X^2))^2}=3\frac{\nu-2}{\nu-4}>3
\end{align}
Or the excess kurtosis:
\begin{align}
\gamma =\kappa-3>0
\end{align}
If $\nu\leq 4$ the fourth moment is simply not defined and you cannot calculate the kurtosis or excess kurtosis. Interpreting something that is not defined is difficult, so don't let it confuse you.
A: "Heaviness of tails" refers to a distribution's ability to produce rare, extreme values ("outliers"). Your model is estimating a distribution with heavy tails. Kurtosis is one measure of heavy tails, but it has limitations in that it can be infinite, as others have noted. The df parameter is also a measure of heaviness of tails; smaller implies heavier. When df=1, you have the famous Cauchy distribution, which is a notoriously heavy tailed distribution.
In my experience using the t distribution, I have been surprised at how small the df parameter is estimated to be. It often seems too small. You should evaluate the confidence interval (asymmetric) for the estimated df parameter to help understand this issue.
To validate the model, you should consider making some probabilistic predictions, such as "When X=0, how often is Y above 4.0 or less than -4?" (Substitute numbers other than 0 and 4 that may make more sense from the subject matter standpoint.) If such probabilities are way out of line with your a priori understanding of the data-generating process, then there is a problem with the model.  You might need a different family of distributions in this case.
A: For student t with $\nu<4$ you end up with very heavy tails. If you measure tail weight with excess kurtosis then it is infinite (undefined) for your parameter. Interpretation wise, it simply means that the extreme observations are much more likely than with Gaussian distribution.
In fact even with $\nu>4$ Student t tails cause serious issues. For instance, if you model $\ln y$ then your $y$ will not have a mean. Consider being warned about Student t residuals.
