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The search for one-step alternatives to the Generalized Method of Moment (GMM) has identified broad classes of potential estimators such as Generalized Empirical Likelihoods (GEL), Empirical Cressie-Read (ECR), Exponentially Tilted Empirical Likelihood (ETEL) and minimum discrepancy (MD) estimators. While Empirical Likelihood (EL) dominates other ECR estimators in terms of higher-order asymptotics, it lacks robustness to model misspecification. ETEL was shown to combine higher-order efficiency and robustness to misspecification, but demands strong moment generating function existence conditions. We show, both theoretically and via simulations, how to achieve the same goal under weaker moment existence conditions, within the class of MD estimators. 

Economic models often depend on quantities that are unobservable, either for privacy reasons or because they are difficult to measure. Examples of such variables include human capital (or ability), personal income, unobserved heterogeneity (such as consumer “types”), et cetera. This situation has historically been handled either by simply using observable imperfect proxies for each of the unobservables, or by assuming that such unobservables satisfy convenient conditional mean or independence assumptions that enable their elimination from the estimation problem. However, thanks to tremendous increases in both the amount of data available and computing power, it has become possible to take full advantage of recent formal methods to infer the statistical properties of unobservable variables from multiple imperfect measurements of them. The general framework used is the concept of measurement systems in which a vector of observed variables is expressed as a (possibly nonlinear or nonparametric) function of a vector of all unobserved variables (including unobserved error terms or “disturbances” that may have nonadditively separable affects). The framework emphasizes important connections with related fields, such as nonlinear panel data, limited dependent variables, game theoretic models, dynamic models, and set identification. This review reports the progress made toward the central question of whether there exist plausible assumptions under which one can identify the joint distribution of the unobservables from the knowledge of the joint distribution of the observables. It also overviews empirical efforts aimed at exploiting such identification results to deliver novel findings that formally account for the unavoidable presence of unobservables. (JEL C30, C55, C57, D12, E21, E23, J24) 

We show that a standard linear triangular two equation system can be point identified, without the use of instruments or any other side information. We find that the only case where the model is not point identified is when a latent variable that causes endogeneity is normally distributed. In this nonidentified case, we derive the sharp identified set. We apply our results to Acemoglu and Johnson’s model of life expectancy and GDP, obtaining point identification and comparable estimates to theirs, without using their (or any other) instrument. 

The idea of summarizing the information contained in a large number of variables by a small number of “factors” or “principal components” has been broadly adopted in statistics. This article introduces a generalization of the widely used principal component analysis (PCA) to nonlinear settings, thus providing a new tool for dimension reduction and exploratory data analysis or representation. The distinguishing features of the method include (i) the ability to always deliver truly independent (instead of merely uncorrelated) factors; (ii) the use of optimal transport theory and Brenier maps to obtain a robust and efficient computational algorithm; (iii) the use of a new multivariate additive entropy decomposition to determine the most informative principal nonlinear components, and (iv) formally nesting PCA as a special case for linear Gaussian factor models. We illustrate the method’s effectiveness in an application to excess bond returns prediction from a large number of macro factors. Supplementary materials for this article are available online. 

Abstract The traditional approach to obtain valid confidence intervals for non-parametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: first, the question of optimal bandwidth selection is no longer well-defined, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a confidence interval is very inefficient. To address these issues, we construct valid confidence intervals that account for the presence of a non-negligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves finding a strict, yet feasible, bound on the bias of a non-parametric estimator. It is well-known that it is not possible to consistently estimate the pointwise bias of an optimal non-parametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone’s bounds on the optimal convergence rates). Nevertheless, we find that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is sufficient to deliver a valid confidence interval whose length decreases at the optimal rate and which does not contradict Stone’s results.