A Cautionary Bedtime Story

Once upon a time, when all the world and you and I were young and beautiful, there lived in the ancient town of Metrika a young boy by the name of Joe.

Happy Birthday, Econometric Society

The Econometric Society was founded 83 years ago today, as a result of a meeting held at the Stalton Hotel in Cleveland, Ohio.

One of my earliest posts was devoted to this aspect of the history of our discipline. If you haven't read it, this would certainly be an appropriate day to do so!

And if you want to look ahead, as well as back, keep in mind that the Econometric Society holds  a World Congress every five years. The 11^th Congress is scheduled for 15 to 21 August 2015, in Montreal, Canada.

See you there!

© 2013, David E. Giles

Statistical Significance - Again

With all of this emphasis on "Big Data", I was pleased to see this post on the Big Data Econometrics blog, today.

When you have a sample that runs to the thousands (billions?), the conventional significance levels of 10%, 5%, 1% are completely inappropriate. You need to be thinking in terms of tiny significance levels.

I discussed this in some detail back in April of 2011, in a post titled, "Drawing Inferences From Very Large Data-Sets". If you're of those (many) applied researchers who uses large cross-sections of data, and then sprinkles the results tables with asterisks to signal "significance" at the 5%, 10% levels, etc., then I urge you read that earlier post.

It's sad to encounter so many papers and seminar presentations in which the results, in reality, are totally insignificant!

© 2013, David E. Giles

Unbiased Estimation of a Standard Deviation

Frequently, we're interested in using sample data to obtain an unbiased estimator of a population variance. We do this by using the sample variance, with the appropriate correction for the degrees of freedom. Similarly, in the context of a linear regression model, we use the sum of the squared OLS residuals, divided by the degrees of freedom, to get an unbiased estimator of the variance of the model's error term.

But what if we want an unbiased estimator of the population standard deviation, rather than the variance?

Solution to Regression Problem

O.K. - you've had long enough to think about that little regression problem I posed the other day. It's time to put you out of your misery!

Here's the problem again, with a solution.

Thought for the Day

As a number of writers have noted previously, sales of Christmas cards Granger-cause Christmas, but they certainly don't cause Christmas!

Best wishes for the holiday season.

© 2013, David E. Giles

A Simple Regression Problem

Here's a regression problem for student readers of this blog.

Suppose that we estimate the following regression model by OLS:

                     y_i = α + β x_i + ε_i .

The model has a single regressor, x, and the point estimate of β turns out to be 10.0.

Now consider the "reverse regression", based on exactly the same data:

                    x_i = a + b y_i + u_i .

What can we say about the value of the OLS point estimate of b?

• It will be 0.1.
• It will be less than or equal to 0.1.
• It will be greater than or equal to 0.1.
• It's impossible to tell from the information supplied.

© 2013, David E. Giles

Thomas Bayes - 250 Years On

Two hundred and fifty years ago today a paper titled, "An Essay Towards Solving a Problem in the Doctrine of Chances", was presented to a meeting of the Royal Statistical Society in London. (Although, see here.)

The presenter - Richard Price. The author - (the late) Reverend Thomas Bayes.

Thus, we received "Bayes' Theorem".

A few days ago, the International Society for Bayesian Analysis held a celebratory conference to honour this momentous occasion in the history of statistical and scientific thinking.

Bayesian thinking has had a significant impact on the field of econometrics. My own Ph.D. dissertation (1975) was in Bayesian econometrics, and I was fortunate enough to have had Arnold Zellner as an external examiner.

I just wish I'd had access to the computational technology that's so freely available today!

© 2013, David E. Giles

More on Student-t Regression Models

My recent post relating to maximum likelihood estimation of non-standard regression models in EViews included the case where the model's errors are independent Student-t distributed. In that example, the degrees of freedom for the Student-t distribution were assumed to be known. There was a good reason for making this assumption, as was spotted by Osman Dogan in his comment on that post.

If we relax this assumption and include the degrees of freedom parameter, v, of the t-distribution as another parameter that has to be estimated, then the likelihood function exhibits some unfortunate characteristics. Specifically, this function becomes unbounded at a boundary of the parameter space. Consequently, maximizing the likelihood function will generally result in us achieving only a local maximum, not a global maximum.

You might ask, "why would this matter?" Well, basically, if you want to be sure that your MLE achieves the good asymptotic properties that motivate us to use it in the first place, then you need to globally maximize the likelihood function.

I discussed this issue in some detail in an earlier post, here.

In the context of the multiple regression model with independent Student-t errors with an unknown degrees of freedom parameter, these issues have been discussed fully by Fernandez and Steel (1999), for example. In particular, those authors show how a Bayesian approach to this estimation problem can overcome the difficulties associated with MLE here.

The problem is very reminiscent of the "incidental parameters" problem that arises widely in statistics, as well as in certain econometric estimation problems. Good examples of this general type of problem in econometrics include "switching regression" models; as well as models of markets that are in disequilibrium; and stochastic frontier production functions.

It's well known that a Bayesian approach is productive in the case of the "incidental parameters" problem, so it shouldn't be too surprising that it's also helpful with the Student-t regression model.

So, if you want to estimate a regression model with independent Student-t errors, and the degrees of freedom parameter associated with that distribution is unknown, then don't use maximum likelihood estimation! The Bayesian estimator discussed by Fernandez and Steel (1999) is one alternative. Pianto (2010) suggests a bootstrap estimator; and another possibility  would be to consider method of moments estimation, which would result in estimates that are at least weakly consistent.

References

Fernandez, C, and M. F. J. Steel, 1999. Multivariate Student-t regression models: Pitfalls and inference. Biometrika, 86, 153-167. (Downloadable version here.)

Pianto, D. M., 2010. A bootstrap estimator for the Student-t regression model.

© 2013, David E. Giles

What is an Econometric Model? Objectivity vs. Reflexivity

In response to my recent post, titled, "The History of Econometrics - An Alternative View", Judea Pearl  sent me a thoughtful and intriguing comment. The comment is posted already, but I think that it deserves more than just being tucked away at the bottom of another post.

So, I am giving Judea's comment additional attention here. I hope that you'll find it interesting, and that it will provoke some much-needed discussion.

Here's Judea's comment in its entirety: