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............

Recursions for the Moments of Some Discrete Distributions

You could say, "Moments maketh the distribution". While that's not quite true, it's pretty darn close.

The moments of a probability distribution provide key information about the underlying random variable's behaviour, and we use these moments for a multitude of purposes. Before proceeding, let's be sure that we're on the same page here.

2019 Econometric Game Results

The Econometric Game is over for another year.

The winning team for 2019 was from the University of Melbourne.

The second and third placed teams were from the Maastricht University and Aarhus University, respectively.

Congratulations to the winning teams, and to all who competed this year!

© 2019, David E. Giles

EViews 11 Now Available

As you'll know already, I'm a big fan of the EViews econometrics package. I always found it to be a terrific, user-friendly, resource when teaching economic statistics and econometrics, and I use it extensively in my own research.

Along with a lot of other EViews users, I recently had the opportunity to "test drive" the beta release of the latest version of this package, EViews 11. 

EViews 11 has now been officially released, and it has some great new features. (Click on the links there to see some really helpful videos.) To see what's now available, check it out here. 

Nice update. Thanks!

© 2019, David E. Giles

SHAZAM!

This past weekend the new movie, Shazam, topped the box-office revenue list with over US$53million - and this is it's first weekend since being released.

Not bad!

Of course, in the Econometrics World, we associate the word, SHAZAM, with Ken White's famous computing package, which has been with us since 1977. 

Ken and I go way back. A few years ago I had a post about the background to the SHAZAM package. In that post I explained what the acronym "SHAZAM" stands for. If you check it out you'll see why it's timely for you to know these important historical facts!

And while you're there, take a look at the links to other tales that illustrate Ken's well-known wry sense of humour.

© 2019, David E. Giles

A Permutation Test Regression Example

In a post last week I talked a bit about Permutation (Randomization) tests, and how they differ from the (classical parametric) testing procedure that we generally use in econometrics. I'm going to assume that you've read that post.

(There may be a snap quiz at some point!)

I promised that I'd provide a regression-based example. After all, the two examples that I went through in that previous post were designed to expose the fundamentals of permutation/randomization testing. They really didn't have much "econometric content".

In what follows I'll use the terms "permutation test" and "randomization test" interchangeably.

What we'll do here is to take a look at a simple regression model and see how we could use a randomization test to see if there is a linear relationship between a regressor variable, x, and the dependent variable, y. Notice that I said a "simple regression" model. That means that there's just the one regressor (apart from an intercept). Multiple regression models raise all sorts of issues for permutation tests, and we'll get to that in due course.

There are several things that we're going to see here:

1. How to construct a randomization test of the hypothesis that the regression slope coefficient is zero.
2. A demonstration that the permutation test is "exact". That it, its significance level is exactly what we assign it to be.
3. A comparison between a permutation test and the usual t-test for this problem.
4. A demonstration that the permutation test remains "exact", even when the regression model is mi-specified by fitting it through the origin.
5. A comparison of the powers of the randomization test and the t-test under this model mis-specification.

What is a Permutation Test?

Permutation tests, which I'll be discussing in this post, aren't that widely used by econometricians. However, they shouldn't be overlooked.

Let's begin with some background discussion to set the scene. This might seem a bit redundant, but it will help us to see how permutation tests differ from the sort of tests that we usually use in econometrics.

Background Motivation

When you took your first course in economic statistics, or econometrics, no doubt you encountered some of the basic concepts associated with testing hypotheses. I'm sure that the first exposure that you had to this was actually in terms of "classical", Neyman-Pearson, testing. 

It probably wasn't described to you in so many words. It would have just been "statistical hypothesis testing". The whole procedure would have been presented, more or less, along the following lines:

Some April Reading for Econometricians

Here are my suggestions for this month:

• Hyndman, R. J., 2019. A brief history of forecasting competitions. Working Paper 03/19, Department of Econometrics and Business Statistics, Monash University.
• Kuffner, T. A. & S. G. Walker, 2019. Why are p-values controversial?. American Statistician, 73, 1-3.
• Sargan, J. D.,, 1958. The estimation of economic relationships using instrumental variables. Econometrica, 26, 393-415. (Read for free online.)  
• Sokal, A. D., 1996. Transgressing the boundaries: Towards a trasnformative hermeneutics of quantum gravity. Social Text, 46/47, 217-252.
• Zeng, G. & Zeng, E., 2019. On the relationship between multicollinearity and separation in logistic regression. Communications in Statistics - Simulation and Computation, published online.
• Zhang, X., S. Paul, & Y-G. Yang, 2019. Small sample bias correction or bias reduction? Communications in Statistics - Simulation and Computation, published online.

© 2019, David E. Giles