Identification in a Binary Choice Panel Data Model with a Predetermined Covariate

TL;DR

This paper investigates the identification of a parameter in a binary choice panel data model with a predetermined covariate, revealing conditions under which the parameter is not point-identified and proposing methods to characterize the identified set.

Contribution

It provides a simple condition for non-identification of the parameter and develops a linear programming approach to characterize the identified set, applicable in models including logit.

Findings

The parameter is often not point-identified in such models.

The identified set can be computed using linear programming.

Numerical examples show the identified set can be informative.

Abstract

We study identification in a binary choice panel data model with a single \emph{predetermined} binary covariate (i.e., a covariate sequentially exogenous conditional on lagged outcomes and covariates). The choice model is indexed by a scalar parameter $\theta$, whereas the distribution of unit-specific heterogeneity, as well as the feedback process that maps lagged outcomes into future covariate realizations, are left unrestricted. We provide a simple condition under which $\theta$ is never point-identified, no matter the number of time periods available. This condition is satisfied in most models, including the logit one. We also characterize the identified set of $\theta$ and show how to compute it using linear programming techniques. While $\theta$ is not generally point-identified, its identified set is informative in the examples we analyze numerically, suggesting that meaningful learning about $\theta$ may be possible even in short panels with feedback. As a complement, we report calculations of identified sets for an average partial effect, and find informative sets in this case as well.