Friday, May 3, 2013

When Will the Adjusted R-Squared Increase?

The coefficient of determination (R2) and t-statistics have been the subjects of two of my posts in recent days (here and here). There's another related result that a lot of students don't seem to get taught. This one is to do with the behaviour of the "adjusted" R2 when variables are added to or deleted from an OLS regression model.

We all know, and it's trivial to prove, that the addition of any variable to such a regression model cannot decrease the R2 value. In fact, R2 will increase with such an addition to the model in general. Conversely, deleting any regressor from an OLS regression model cannot increase (and will generally reduce) the value of R2.

Mark Thoma on "Replication"

Yesterday, in his Economist's View blog, Mark Thoma discussed the importance of replicating results in empirical economics. He's absolutely right, of course.

I'll leave you to read what had to say, but I especially liked his closing passage:
"One place where replication occurs regularly is assignments in graduate classes. I routinely ask students to replicate papers as part of their coursework. Even if they don't find explicit errors (and most of the time they don't), it almost always raises good questions about the research (why this choice, this model, what if you relax this assumption, there's a better way to do this,here's the next question to ask, etc., etc.). So replication does occur routinely in economics, and it is very valuable, but it is not a formal part of the profession the way it should be, and much of the replication is done by people (students) who generally assume that if they can't replicate something, it is probably their error. We have a lot of work to do on the replication front, and I want to encourage efforts like this."
At least one of my colleagues also assigns replication exercises in this way, and I really should do the same. Fortunately, more journals are either recommending or requiring that data-sets be made available as a condition of publication. The Journal of Applied Econometrics is one such journal, and we've recently been pushing in that direction with the Journal of International Trade & Economic Development.

This should become part of our culture.

© 2013, David E. Giles

When Can Regression Coefficients Change Sign?

Let's suppose that you've been running regressions happily all morning. It's sunny day, but what could be better than enjoying some honest-to-goodness econometrics? Suddenly, you notice that one of the estimated coefficients in your model has a sign that's the opposite to what you were expecting (from your vast knowledge of the underlying economics). Shock! Horror!

Well. it's really good that you're on the look-out for that sort of thing. Congratulations! However, something has to be done about this problem.

Being young, with good eyesight, you also happen to spot something else that's interesting. One of the other estimated coefficients has a very low t-statistic. You have a brilliant idea! If you delete the variable associated with the very small t-value, maybe the "wrong" sign on the first coefficient will be reversed. Is this possible?