The key assumption of the differences-in-differences approach in the event-study design is that untreated potential outcome differences are mean independent of treatment timing: the parallel trend assumption. In this paper, we relax the parallel trend assumption by assuming a latent type variable and developing a type-specific parallel trend assumption. With a finite support assumption on the latent type variable, we show that an extremum classifier consistently estimates the type assignment. Based on the classification result, we propose a type-specific diff-in-diff estimator for type-specific CATT. By estimating the CATT with regard to the latent type, we study heterogeneity in treatment effect, in addition to heterogeneity in baseline outcomes.

I develop a multilevel model for empirical contexts where each individual belongs to a cluster and a treatment is endogenously assigned at the cluster level. When an explanatory variable of interest is assigned at the cluster level, e.g. clustered treatment, its effect on cluster-level or individual-level outcome cannot be identified in a model with fully flexible cluster heterogeneity. To put restrictions on cluster heterogeneity, I assume that the cluster-level heterogeneity is a function of the cluster-level distribution of individual-level characteristics within each cluster. Since the distribution function is a high-dimensional object for large clusters, two functional analysis methods with dimension reduction properties are used: K-means clustering and functional PCA.