We consider a dynamic panel data model that accounts for a latent group structure across individuals which is constant over time. Differently from the previous literature, we adopt a structural modeling that assumes that the individual effects are generated from a finite mixture with an unknown number of components and unknown parameters for each components. We first establish identification of this model. Then, we specify a prior for the number of components, the parameters of the mixture as well as for the coefficients of the dynamic and exogenous covariates. This extends the mixture of finite mixtures model to panel data settings. We establish asymptotic frequentist properties for the posterior of the parameters of interest as well as for the number of components. A Monte Carlo exercise illustrates finite sample properties.

We study inference on parameters of the form φ(θ0), where φ is a known directionally differentiable transformation and θ0 is an unknown parameter. We focus on settings, where θ0 is an unknown function estimated using some nonparametric estimator ˆθn. As many nonparametric estimators do not converge in distribution, existing extensions to the Delta method are not applicable in our setting. We propose to use strong approximations to the distribution of ˆθn as an alternative concept to convergence in distribution. Further, we present a notion of directional differentiability which is sufficiently flexible to handle the irregularity of nonparametric estimators. These concepts enable us to derive a new Delta method which approximates the distribution of the plug-in estimator φ(ˆθn). Since these distributional approximations are rarely pivotal, we suggest a simulation-based estimator and provide conditions for its consistency. Confidence intervals based on this estimator are shown to provide local size control under conditions on the directional derivative of φ. We illustrate the applicability of our results in three examples and study its finite sample performance in a simulation study.

This paper presents identification and estimation results in the presence of latent group-specific heterogeneity in short binary outcome panels when covariates are available. Specifically, we assume that each unit can be assigned to a time-constant group and model the joint distribution of the binary outcomes conditional on the covariates across all time periods as a mixture model. In our baseline model, we allow the component weights to be fully flexible functions of all covariates over time, while we assume that the binary outcomes are independent over time given the covariates, and the component distributions depend on the contemporaneous covariates only. We present nonparametric identification results for the component weights and distributions under weak conditions. For instance, when the component weights are positive on the entire covariate support and the component distributions exhibit sufficient variation at a single point in the covariate space, the component distributions are nonparametrically identified on the entire support of the covariates. We leverage an inherent exclusion restriction of the model to solve the intra-component label switching problem without further assumptions. In our baseline model, the number of time periods needed for identification depends on the number of groups in a less stringent way than in a setting without covariates. Specifically, identification is possible with as few as two time periods. This dependence changes in a dynamic panel or under additional exclusion restrictions. Our identification results carry over to settings with continuous outcomes or discrete outcomes with support larger than 2. Additionally, we develop a semiparametric estimator and study its asymptotic properties.