Simple approaches to nonlinear difference-in-differences with panel data

Abstract

Summary I derive simple, flexible strategies for difference-in-differences settings where the nature of the response variable may warrant a nonlinear model. I allow for general 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 in the linear exponential family extend pooled ordinary least squares estimation of linear models. By using the conditional mean associated with the canonical link function, imputation and pooling across the entire sample produce identical estimates. Generally, pooled estimation 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.