Identifying Finite Mixtures in Econometric Models

Abstract

Models with unobserved heterogeneity, hidden Markov chains, and many other econometric models are based on mixtures of distributions. It seems crucial to have a general approach to identify them nonparametrically. Yet the literature so far only contains isolated examples, applied to specific models. We derive the identifying implications of a conditional independence assumption in finite mixture models. It applies for instance to models with unobserved heterogeneity, regime switching models, and models with mismeasured discrete regressors. Under this assumption, we derive sharp bounds on the mixture weights and components. For models with two mixture components, we show that if in addition the components behave differently in the tails, then components and weights are fully non-parametrically identified. We apply our findings to the nonparametric identification and estimation of outcome distributions with a misclassified binary regressor. This provides a new simple estimator that does not require instrumental variables, auxiliary data, symmet- ric error distributions or other shape restrictions.