Finally, let’s return to the competing theories of district spending behavior discussed earlier. As it happens, the control variables for both the “demand-driven” thesis (aggregate income per capita) and the public choice thesis (aggregate income per public school pupil squared) are statistically significant. But their explanatory power differs dramatically.

A common way of assessing the importance of a variable’s contribution to a statistical model is to drop it from the equation and see how much the “R-squared” value drops as a result.[*] The resulting difference in the R-squared value is sometimes called “Darlington’s usefulness statistic.”

When income per capita is dropped from the model, the R-squared value falls by just over 1 point. When income per public school pupil squared is dropped, R-squared falls by over 18 points. In fact, the public choice control for bureaucratic behavior has by far the largest Darlington usefulness statistic of any variable in the model other than the “year2” dummy variable.[†] For comparison, the Darlington value for the checkmark function of district size — the independent variable this paper was written to investigate — is only 2.4. Further evidence of the greater explanatory power of the public choice variable appears in “Appendix B: Postestimation Diagnostics.”


[*] R-squared represents the percentage of the variance in the dependent variable — in this case, a district’s per-pupil operating spending — that is explained by the model.

[†] Note, however, that the year dummy variables are controlling for period effects, including inflation, and the effects on spending due to inflation are essentially imaginary (rising spending from one year to the next in current dollars, due to inflation, does not represent a change in real spending).