Global estimation of finite mixture and misclassification models with an application to multiple equilibria

We show that the identification results of finite mixture and misclassification models are equivalent in a widely-used scenario except an extra ordering assumption. In the misclassification model, an ordering condition is imposed to pin down the precise values of the latent variable, which are also of researchers' interests and need to be identified. In contrast, the identification of finite mixture models is usually up to permutations of a latent index. This local identification is satisfactory because the latent index does not convey any economic meaning. However, reaching global identifition is important for estimation, especially, when researchers use bootstrap to estimate standard errors, which may be wrong without a global estimator. We provide a theoretical framework and Monte Carlo evidences to show that imposing an ordering condition to achieve a global estimator innocuously improves the estimation of fine mixture models. As a natural application, we show that games with multiple equilibria fit in our framework and the global estimator with ordering assumptions provides more reliable estimates.