Simple Approaches to Nonlinear Difference-in-Differences with Panel Data

Abstract

I derive simple, flexible strategies for difference-in-differences settings where the nature of the response variable may warrant a nonlinear model. In addition to covering the case of common treatment timing, I allow for staggered interventions, with and without covariates. Under an index version of parallel trends, I show that average treatment effects on the treated (ATTs) are identified for each cohort and calendar time period in which a cohort was subjected to the intervention. The pooled quasi-maximum likelihood estimators (QMLEs) in the linear exponential family (LEF) extend the pooled ordinary least squares (POLS) estimation in Wooldridge (2021). By using the conditional mean associated with the canonical link function, imputation and estimation pooled across the entire sample produce identical results. Moreover, using the canonical link results in very simple computation of the ATTs and their standard errors. The leading cases are a logit functional form for binary and fractional outcomes, combined with the Bernoulli quasi-log likelihood (QLL), and an exponential mean combined with the Poisson QLL. A small simulation study shows the estimators work well when the mean function is correctly specified; they also have some resiliency to misspecification.

Keywords: difference-in-differences, staggered intervention, nonlinear model, logit, poisson regression

JEL Classification: C23, C54