Some Properties of Non-linear Least Squares

You probably know that when we have a regression model that is non-linear in the parameters, the Non-Linear Least Squares (NLLS) estimator is generally biased, but it's weakly consistent. This is the case even if the model has non-random regressors and an additive error term that satisfies all of the usual assumptions.

In addition, even if the model’s errors are normally distributed, the NLLS estimator will have a sampling distribution that is non-normal in finite samples, and the usual t-statistics will not be Student-t distributed in finite samples.

In this post I'll illustrate these, and some other results, by using a simple Monte Carlo experiment.

Central Limit Theorems

When we first encounter asymptotic (large sample) theory in econometrics, one of the most important results that we learn about is the Central Limit Theorem.  Loosely speaking we learn that if we aggregate together enough values that are sampled randomly from the same distribution, with a finite mean and variance, then this aggregate starts to behave as if it is normally distributed.

However, too few courses make it clear that this "classical" central limit theorem is just one of several such results. The one that assumes independently and identically distributed values is actually the Lindeberg-Lévy Central Limit Theorem. There are other, related, results that deal with less restrictive situations.

Viren Srivastava

Recently, a reader of this blog asked I could provide some information about the late V.K. Srivastava, and the substantial contributions that he made to econometrics and to statistics generally.

I'm more than happy to oblige, as Virendra (Viren) was a good friend of mine, a treasured co-author, and a very caring and humble individual.

Jobs for Econometricians

My impression is that there is a strong  international market for economists who have strong skills in econometrics. I'm not talking just jobs in the academic community, but also about positions in the private, public, and non-profit sectors too.

 

This blog has a page that list a very small selection of such jobs. This list has never been meant to be exhaustive. That's not what this blog is about. Rather, the jobs listed on that page are meant to be illustrative of some of the various jobs that are available to econometricians.

 

If you're looking seriously for an academic position, especially at the entry level, then the obvious place to start is Job Openings for Economists (JOE). This is sponsored by the American Economics Association, but handles jobs internationally. Although the focus is on academic positions, jobs in other sectors appear in JOE too.

 

Another website that may interest you is econometricsjobs.com. This is a commercial site that lists positions specific to econometrics. It also has international coverage, and covers all sectors of the workforce. Just browsing some of the jobs that are advertised there may broaden your perception of the opportunities that are available to econometricians.

 

There are other sites too, of course. Perhaps some of these will get mentioned in comments to this post. 

© 2012, David E. Giles

Mathgen

H/T to my colleague, Martin Farnham, for drawing my attention to Mathgen.

 

Thanks to Nate Eldridge, a mathematician at Cornell University, who blogs at That's Mathematics!, you can randomly generate your own mathematics research paper!

 

In fact, a Mathgen-generated was recently accepted for publication at one of those pseudo-journals that seem to have sprouted with a vengeance of late. If you weren't convinced already that these publishing outlets should be avoided like the plague, this ought to do it for you!

 

Just for funzies, I decided to solicit Mathgen's assistance in writing my own paper. It took just a few seconds, and you can read it here. Constructive comments are welcomed, of course. Just don't ask me what the title means.

I have a feeling that this is going to be a particularly productive weekend!

 

(As Martin suggested to me, this is every journal editor's new nightmare!)

 

© 2012, David E. Giles

Let's be Consistent

One of the standard, large-sample, properties that we hope our estimators will possess is "consistency". Indeed, most of us take the position that if an estimator isn't consistent, then we should probably throw it away and look for one that is!

When you're talking about the consistency of an estimator, it's a really good idea to be quite clear regarding the precise type of consistency you have in mind - especially if you're talking to a statistician! For example, there's "weak consistency", "strong consistency", "mean square consistency", and "Fisher consistency", at least some of which you'll undoubtedly encounter from time to time as an econometrician.

Some Historical Links

You've probably noticed that some of my posts are essentially pieces that focus on some aspect of the history of econometrics, and/or the history of statistics.  I certainly have a bit of an interest in these topics, and I also find that it's helpful to inject a bit of historical content when I'm teaching. 

It doesn't necessarily have to be very much - just something interesting to make the name of the econometrician in question, or the origin of a concept a bit more memorable. Or perhaps some historical context that's intended to clarify why the literature took a certain turn at a certain time.

It's both interesting and enlightening to know something about where your discipline came from, how it evolved over time, and who the players were. Some of them were really interesting people!

What I Learned Last Week

Somewhat to my surprise, last month I got a great response to my post, "My Must-Read List" (HT's to Mark Thoma & Tyler Cowen). This past week I learned a lot by reading some terrific new papers on a variety of econometrics topics. Here they are, with some commentary, and in no particular order:

Degrees of Freedom in Regression

Yesterday, one of the students from my introductory grad. econometrics class was asking me for more explanation about the connection between the "degrees of freedom" associated with the OLS regression residuals, and the rank of a certain matrix. I decided to out together a quick handout to do justice to her question, and it occurred to me that this handout might also be of interest to a wider group of student readers.

So, here's what I wrote.

How Good is Your Random Number Generator?

Simulation methods, including Monte Carlo simulation and various forms of the bootstrap, are widely used by econometricians. We use these tools to learn about the sampling distributions of our estimators and tests, especially in situations where a purely analytic approach is technically difficult.

For example, sometimes we're able to appeal to standard asymptotic (large sample) results - such as the central limit theorems, and the laws of large numbers - to figure out how good our inferences will be if the sample size is very large. However, when it comes to the question of how good they are when the sample size is quite small, the answer may not be so easily established.

In addition, when we come up with a new theoretical result in econometrics, most of us take the precaution of also simulating the result - as check on its accuracy.

Monte Carlo and bootstrap methods rely critically on our ability to generate "pseudo"-random numbers that have the characteristics that we ascribe to them. How often have you actually checked  if the random number generators in your favourite econometrics package produce values that are "random", and follow the distribution that you've asked for? Probably not often enough!

I follow John Cook's blog, The Endeavour. A couple of years ago he had a nice post titled, "How to test a random number generator". In that post, he links to a chapter of the same title that he wrote for the book, Beautiful Testing (edited by Tim Riley and Adam Goucher).

John's chapter is a short, but very valuable read, and I recommend it strongly.

© 2012, David E. Giles