Empirical Legal Studies

"Paired" v. "Unpaired" T-Test

Efforts to compare the means from two sample populations with a standard t-test invariably bump into a "paired" versus and "unpaired" decision. Whether a "paired" or "unpaired" comparison is appropriate pivots on a subtle distinction that can be difficult to describe. A recent Statalist discussion might provide a helpful way to think about this distinction.

The discussion (here) began with the following (edited) question: "If there is economic growth data about two countries over the same thirty years, should it be considered as paired or unpaired data for two sample t-test?"

The various responses included the following: "For unpaired tests imagine 30 men and 30 women completely unrelated and compare their heights. You are comparing the mean male height and mean female height and the context is the variability of male heights and of female heights. For paired tests imagine 30 heterosexual married couples and compare height for each husband with each wife."

With this in mind, the answer to the question posed is: "Your panel example is still paired because there is pairing by years."

Log Transformations & the "Zeros Problem"

Strategies to address normal distribution assumption concerns can quickly lead to the log transformation of variables. However, what to do with "zeros" in a variable requiring log transformation persists as an under-studied issue. And one well-known example, punitive damage awards, makes clear that such "zeros problems" are not uncommon in many empirical legal research projects.

In Dealing With the Log of Zero in Regression Models, the authors, Christophe Bellego (Center for Research in Economics and Statistics) and Louis-Danile Pape (Center for Research in Economics and Statistics), engage with this nettlesome issue head-on. The brief (15 pp.) paper's abstract follows.

"Log-linear and log-log regressions are one of the most used statistical model. However, handling zeros in the dependent and independent variable has remained obscure despite the prevalence of the situation. In this paper, we discuss how to deal with this issue. We show that using Pseudo-Poisson Maximum Likelihood (PPML) is a good practice compared to other approximate solutions. We then introduce a new complementary solution to deal with zeros consisting in adding a positive value specific to each observation that avoids some numerical issues faced by the former."

Ready-To-Use Data (and Codebooks) for Law & Education Research

One thing that the U.S. Department of Education does reasonably well is gather data--and make them publicly available. And as the number and complexity of law and education issues continue to increase over time, greater access to relevant data becomes even more important. To this end, the Dept. of Education's National Center for Education Statistics ("NCES") operates a landing spot (here) for direct access to ready-to-use public use data sets and codebooks. Users can select from an array of standard formats (Stata, SPSS, SAS, R, etc.) and tailor requests.

Instrumental Variables Regression--A Helpful Intro

For a brief (5+ min) introduction to--or a very quick refresher on--the basics of instrumental variables regression, including some post-estimation commands, one could do far worse than this Stata video. The video walks viewers through the basics on how to fit an instrumental variables model for endogenous covariates (using the "ivregress" command).