Estimating Grouped Data Models with a Binary-Dependent Variable and Fixed Effects via a Logit versus a Linear Probability Model: The Impact of Dropped Units

This letter deals with a very simple question: if we have grouped data with a binary-dependent variable and want to include fixed effects in the specification, can we meaningfully compare results using a linear model to those estimated with a logit? The reason to doubt such a comparison is that the linear specification appears to keep all observations, whereas the logit drops the groups where the dependent variable is either all zeros or all ones. This letter demonstrates that a linear specification averages the estimates for all the homogeneous outcome groups (which, by definition, all have slope coefficients of zero) with the slope coefficients for the groups with a mix of zeros and ones. The correct comparison of the linear to logit form is to only look at groups with some variation in the dependent variable. Researchers using the linear specification are urged to report results for all groups and for the subset of groups where the dependent variable varies. The interpretation of the difference between these two results depends upon assumptions which cannot be empirically assessed.