Fixed Effects Logit Estimator and the Incidental Parameter Problem

Fixed effects (FE) regressions are a useful tool for controlling for time-invariant factors in a regression specification.  When using a linear OLS model, FE represent the average value of the dependent variable for that individual after controlling for covariates.  Estimating a fixed effects model for non-linear regressions, however, can be problematic.

For instance, if you try to estimate the fixed effects coefficients in a probit model, you will introduce an incidental parameters problem.  Assume that the panel data has N individuals over T time periods.  If T is fixed, as N grows large (i.e., N→∞) your covariate estimates (β) become biased.  This occurs because the number of “nuisance parameters” grow quickly as N increases.

There does exist a “fixed effects logit estimator”, but this estimator does not actually use a fixed effects method.  Rather it is a conditional maximum likelihood estimator (cMLE).  In the two period model, it conditions on the fact that the event occurred in one or the other time period.

  • Pr(yi1=1|yi1+yi2=1) = 1-F(Δxβ)
  • Pr(yi2=1|yi1+yi2=1) = F(Δxβ)

“The conditional-likelihood estimator is thus equivalent to a logit estimator of the dependent variable 1(Δy=1) on the independent variables Δx for the subsample of observations satisfying yi1 + yi2=1.”

Abrevaya (1996) provides an example of how this bias can occur in a two period example and explains the conditional logit model in more detail.

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