A Warning Concerning Random Effects and Random Coefficients in Logistic Regression Models for Binary Data

For binary panel data, the introduction of a random respondent effect in a logistic regression model is a useful way of taking respondent heterogeneity into account. More generally, logistic regression models with random coefficients can be used if not only the intercept, but also the coefficients to explanatory variables can be expected to vary from respondent to respondent. However, there arc some identifiability problems with these models in the special case where respondents arc observed only once. A clarification of these matters can be obtained by studying the probit-lincar model rather than the logit-lincar model. In practice this change of link function makes very little difference. But the advantage of the probit models is that the identifiability problems - which in the logit models with normal random effects merely result in numerically unstable solutions to the likelihood equations - correspond to mathematically exact overparamctrizations in the probit-lincar models.