Zero-One Matrices

When we're learning the basics of least squares regression analysis, one of the topics that we invariably encounter is the consequences of model mis-specification. In particular, we're taught that omitting relevant regress from the model renders the OLS estimator biased and inconsistent, although its precision is improved. On the other hand, including extraneous regressors simply reduces the efficiency of the OLS estimator of the coefficient vector. That estimator is still unbiased (and consistent) in this case.

These results are just special cases of those associated with imposing false restrictions on the parameter space, or failing to impose valid restrictions. So, once these more general results have been covered there's really no need to treat the "omitted regressors" and "extraneous regressors" situations as a separate matter.

However, usually they are dealt with as a distinct topic. What I find interesting, and what I want to focus on here, is the way in which the unbiasedness of OLS can be demonstrated in the context of irrelevant regressors. There's an easy way to get this result, and there's a more tedious proof. Let's begin by looking at the easy way.

Congratulations, Sir Richard!

Well-known microeconometrician, Richard Blundell, has been knighted in the New Year's Honours list. Sir Richard is the Ricardo Professor of Political Economy at University College London, and Research Director for the Institute for Fiscal Studies.

Knighted for his services to economics and social science, Sir Richard has previously served as co-editor of both Econometrica and Journal of Econometrics, and his many other achievement and awards are listed here.

I'm sure that all econometricians will be delighted by this recognition of Sir Richard's contributions, and offer him their sincere congratulations!

© 2014, David E. Giles

My Top 5 For 2013

Everyone seems to be doing it at this time of the year. So, here are the five most popular new posts on this blog in 2013:

1. Econometrics and "Big Data"
2. Ten Things for Applied Econometricians to Keep in Mind
3. ARDL Models - Part II - Bounds Tests
4. The Bootstrap - A Non-Technical Introduction
5. ARDL Models - Part I

Thanks for reading, and for your comments.

Happy New Year!


© 2013, David E. Giles

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