Monday, April 29, 2019

Recursions for the Moments of Some Continuous Distributions

This post follows on from my recent one, Recursions for the Moments of Some Discrete Distributions. I'm going to assume that you've read the previous post, so this one will be shorter. 

What I'll be discussing here are some useful recursion formulae for computing the moments of a number of continuous distributions that are widely used in econometrics. The coverage won't be exhaustive, by any means. I provide some motivation for looking at formulae such as these in the previous post, so I won't repeat it here. 

When we deal with the Normal distribution, below, we'll make explicit use of Stein's Lemma. Several of the other results are derived (behind the scenes) by using a very similar approach. So, let's begin by stating this Lemma.

Stein's Lemma (Stein, 1973):

"If  X ~ N[θ , σ2], and if g(.) is a differentiable function such that E|g'(X)| is finite, then 

                            E[g(X)(X - θ)] = σ2 E[g'(X)]."

It's worth noting that although this lemma relates to a single Normal random variable, in the bivariate Normal case the lemma generalizes to:

"If  X and Y follow a bivariate Normal distribution, and if g(.) is a differentiable function such that E|g'(Y)| is finite, then 

                            Cov.[g(Y )X] = Cov.(X , Y) E[g'(Y)]."

In this latter form, the lemma is useful in asset pricing models.

There are extensions of Stein's Lemma to a broader class univariate and multivariate distributions. For example, see Alghalith (undated), and Landsman et al. (2013), and the references in those papers. Generally, if a distribution belongs to an exponential family, then recursions for its moments can be obtained quite easily.

Now, let's get down to business............