Reporting an R-Squared Measure for Count Data Models

This post was prompted by an email query that I received some time ago from a reader of this blog. I thought that a more "expansive" response might be of interest to other readers............

In spite of its many limitations, it's standard practice to include the value of the coefficient of determination (R²) - or its "adjusted" counterpart - when reporting the results of a least squares regression. Personally, I think that R² is one of the least important statistics to include in our results, but we all do it. (See this previous post.)

If the regression model in question is linear (in the parameters) and includes an intercept, and if the parameters are estimated by Ordinary Least Squares (OLS), then R² has a number of well-known properties. These include:

1. 0 ≤ R² ≤ 1.
2. The value of R² cannot decrease if we add regressors to the model.
3. The value of R² is the same, whether we define this measure as the ratio of the "explained sum of squares" to the "total sum of squares" (R_E²); or as one minus the ratio of the "residual sum of squares" to the "total sum of squares" (R_R²).
4. There is a correspondence between R² and a significance test on all slope parameters; and there is a correspondence between changes in (the adjusted) R² as regressors are added, and significance tests on the added regressors' coefficients.   (See here and here.)
5. R² has an interpretation in terms of information content of the data.  
6. R² is the square of the (Pearson) correlation (R_C²) between actual and "fitted" values of the model's dependent variable. 

However, as soon as we're dealing with a model that excludes an intercept or is non-linear in the parameters, or we use an estimator other than OLS, none of the above properties are guaranteed.

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.

September Reading List

In North America, Labo(u)r Day weekend is upon us. The end of summer. Back to school. Last chance to get some pre-class reading done!

• Blackburn, M. L., 2014. The relative performance of Poisson and negative binomial regression estimators. Oxford Bulletin of Economics and Statistics, in press.
• Giannone, D., M. Lenza, and G. E. Primiceri, 2014. Prior selection for vector autoregressions. Review of Economics and Statistics, in press.
• Gulesserian, S. G. and M. Kejriwal, 2014. On the power of bootstrap tests for stationarity: A Monte Carlo comparison. Empirical Economics, 46, 973-998.
• Elliot, G. and A. Timmerman, 2008. Economic forecasting. Journal of Economic Literature, 46, 3-56.
• Kiviet, J. F., 1986, On the rigour of some misspecification tests for modelling dynamic relationships. Review of Economic Studies, 53, 241-261.
• Otto, G. D. and G. M. Voss, 2014. Flexible inflation forecast targeting: Evidence from Canada. Canadian Journal of Economics, 47, 398-421. 

© 2014, David E. Giles

Modelling Olympic Medal Wins

With the Sochi Winter Olympics now well underway, I was reminded of the empirical literature that has attempted to model the number of medals that different countries win.

Back in 2006, one of our students, Glen Roberts, wrote an excellent paper, titled "Accounting for Achievement in Athens: A Count Data Analysis of National Olympic Performance", on this topic. Glen's work was based on a term project that he undertook for my ECON 546, "Themes in Econometrics" course. This is an elective course for M.A. students, and it emphasises the thematic content of econometric methods - MLE, IV/GMM, Bayesian inference, etc.

Glen found that the empirical analyses of Olympic medal wins largely ignored the "count data" aspect of the problem. You can find plenty of references in Glen's paper. He then set about rectifying this situation, as the abstract to his paper describes:

Variance Estimators That Minimize MSE

In this post I'm going to look at alternative estimators for the variance of a population. The following discussion builds on a recent post, and once again it's really directed at students. Well, for the most part.

Actually, some of the results relating to populations that are non-Normal probably won't be familiar to a lot of readers. In fact, I can't think of a reference for where these results have been assembled in this way previously. So, I think there's some novelty here. But we'll get to that in due course.

I can just imagine you smacking your lips in anticipation!

What's the Variance of a Sample Variance?

This post is really pitched at students who are taking a course or two in introductory economic statistics. It relates to a couple of estimators of the variance of a population that we all meet in such courses - plus another one that you might not have met. In addition, I'll be emphasising the fact that some "standard" results depend crucially on certain assumptions. Not surprisingly - but  not always made clear by instructors and text books.