Conditional Logit with one Binary Covariate: Link between the Static and Dynamic Cases

Disentangling state dependence from unobserved heterogeneity is a common issue in economics. It arises for instance when studying transitions between different states on the labor market. When the outcome variable is binary, one of the usual strategies consists in using a conditional logit model with an appropriate conditioning suitable for a dynamic framework. Although static conditional logit procedures are widely available, these procedures cannot be used directly in a dynamic framework. Indeed, it is inappropriate to use them with a lag dependent variable in the list of regressors. Moreover, reprogramming this kind of procedures in a dynamic framework can prove quite cumbersome because the likelihood can have a very high number of terms when the number of periods increases. Here, we consider the case of a conditional logit model with one binary regressor which can be either exogenous or the lagged dependent variable itself. We provide closed forms for the conditional likelihoods in both cases and show the link between them. These results show that in order to evaluate a conditional logit model with one lag of state dependence and no other covariate, it is possible to simply generate a two variable dataset and use standard procedures originally intended for models without state dependence. Moreover, the closed forms help reduce the computational burden even in the static case in which preimplemented procedures usually exist.