Estimating grouped data models with a binary dependent variable and fixed effects: What are the issues

TL;DR

This paper compares OLS and fixed effects logit models for binary grouped data, showing that fixed effects logit methods generally outperform OLS, especially for estimating marginal effects in small groups or many groups.

Contribution

It demonstrates that fixed effects logit models, particularly the Chamberlain conditional logit, are superior to OLS for binary data with fixed effects, clarifying issues related to incidental parameters and marginal effects estimation.

Findings

Fixed effects logit models outperform OLS in estimating binary grouped data.

Chamberlain conditional logit is as good as or better than simple fixed effects logit.

Using conditional logit estimates to constrain fixed effects logit improves marginal effects estimation.

Abstract

This article deals with asimple issue: if we have grouped data with a binary dependent variable and want to include fixed effects (group specific intercepts) in the specification, is Ordinary Least Squares (OLS) in any way superior to a (conditional) logit form? In particular, what are the consequences of using OLS instead of a fixed effects logit model with respect to the latter dropping all units which show no variability in the dependent variable while the former allows for estimation using all units. First, we show that the discussion of fthe incidental parameters problem is based on an assumption about the kinds of data being studied; for what appears to be the common use of fixed effect models in political science the incidental parameters issue is illusory. Turning to linear models, we see that OLS yields a linear combination of the estimates for the units with and without variation in the dependent variable, and so the coefficient estimates must be carefully interpreted. The article then compares two methods of estimating logit models with fixed effects, and shows that the Chamberlain conditional logit is as good as or better than a logit analysis which simply includes group specific intercepts (even though the conditional logit technique was designed to deal with the incidental parameters problem!). Related to this, the article discusses the estimation of marginal effects using both OLS and logit. While it appears that a form of logit with fixed effects can be used to estimate marginal effects, this method can be improved by starting with conditional logit and then using the those parameter estimates to constrain the logit with fixed effects model. This method produces estimates of sample average marginal effects that are at least as good as OLS, and much better when group size is small or the number of groups is large. .