Partial identification of finite mixtures in econometric models

Abstract

We consider partial identification of finite mixture models in the presence of an observable source of variation in the mixture weights that leaves component distributions unchanged, as is the case in large classes of econometric models. We first show that when the number J of component distributions is known a priori, the family of mixture models compatible with the data is a subset of a J(J − 1)dimensional space. When the outcome variable is continuous, this subset is defined by linear constraints, which we characterize exactly. Our identifying assumption has testable implications, which we spell out for J = 2. We also extend our results to the case when the analyst does not know the true number of component distributions and to models with discrete outcomes.

Inference on Two-Component Mixtures Under Tail Restrictions

“…1 Assumption 1 (Mixture with exclusion). F(y|x) ≡ P(Y ≤ y|X = x) decomposes as the two-component mixture The assumption that the component distributions do not depend on X embodies our exclusion restriction; see also Henry et al (2014).…”

“…The exclusion restriction is also implied by the conditional-independence restriction that underlies the results of Hall and Zhou (2003) and others on multivariate mixtures. Henry et al (2014) have shown that our exclusion restriction implies that both the mixing proportions and the component distributions lie in a nontrivial set. However, they only proved partial identification, and they did not discuss inference.…”

“…Like Henry et al (2014), we assume the existence of a source of variation that shifts the mixing proportions but leaves the component distributions unchanged. Such an assumption is natural in several important applications, such as measurement-error models (Mahajan, 2006), for example.…”

“…A nonexhaustive list of applications includes models with discrete unobserved heterogeneity, hidden Markov chains, and models with mismeasured discrete variables; see Henry, Kitamura, and Salanié (2014) for a more extensive discussion of applications. Until recently, the literature on nonparametric identification of mixture models was sparse.…”

Nonparametric spectral-based estimation of latent structures

“…Their approach can be applied to a more general class of latent structures that feature some form of conditional independence, such as hidden Markov models with finite state spaces (see Petrie [1969] for seminal work on this) and to random-graph models. We note that, although the availability of two measurements can suffice in problems featuring additive structures, the work of Henry, Kitamura and Salanié [2013] shows that two measurements will only deliver set-identification of parameters in more general latent-structure models. Li and Vuong [1998], Bordes, Mottelet and Vandekerkhove [2006], and Gassiat and Rousseau [2013], among others, present results for additive models.…”