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We consider the identification of and inference on a partially linear model, when the outcome of interest and some of the covariates are observed in two different datasets that cannot be linked. This type of data combination problem arises very frequently in empirical microeconomics. Using recent tools from optimal transport theory, we derive a constructive characterization of the sharp identified set. We then build on this result and develop a novel inference method that exploits the specific geometric properties of the identified set. Our method exhibits good performances in finite samples, while remaining very tractable. Finally, we apply our methodology to study intergenerational income mobility over the period 1850-1930 in the United States. Our method allows to relax the exclusion restrictions used in earlier work while delivering confidence regions that are informative.

In this paper, we build a new test of rational expectations based on the marginal distributions of realizations and subjective beliefs. This test is widely applicable, including in the common situation where realizations and beliefs are observed in two dierent datasets that cannot be matched. We show that whether one can rationalize rational expectations is equivalent to the distribution of realizations being a mean-preserving spread of the distribution of beliefs. The null hypothesis can then be rewritten as a system of many moment inequality and equality constraints, for which tests have been recently developed in the literature. The test is robust to measurement errors under some restrictions and can be extended to account for aggregate shocks. Finally, we apply our methodology to test for rational expectations about future earnings. While individuals tend to be right on average about their future earnings, our test strongly rejects rational expectations

We study the informational content of factor structures in discrete triangular systems. Factor structures have been employed in a variety of settings in cross sectional and panel data models, and in this paper we formally quantify their identifying power in a bivariate system often employed in the treatment effects literature. Our main findings are that imposing a factor structure yields point identification of parameters of interest, such as the coefficient associated with the endogenous regressor in the outcome equation, under weaker assumptions than usually required in these models. In particular, we show that a "non-standard" exclusion restriction that requires an explanatory variable in the outcome equation to be excluded from the treatment equation is no longer necessary for identification, even in cases where all of the regressors from the outcome equation are discrete. We also establish identification of the coefficient of the endogenous regressor in models with more general factor structures, in situations where one has access to at least two continuous measurements of the common factor.

Using data from a Canadian field experiment on the financial barriers to higher education, we estimate the distribution of the value of financial aid for prospective students. Our results point out that a considerable share of prospective students are affected by credit constraints. We find that most of the individuals are willing to pay a sizable interest premium above the prevailing market rate for the option to take up a loan, with a median interest rate wedge equal to 6.8 percentage points for a $1,000 loan. The willingness-to-pay for financial aid is highly heterogeneous across students, with preferences and in particular discount factors, playing a key role in accounting for this variation.

This article presents the eqregsel command for implementing the estimation and bootstrap inference of sample selection models via extremal quantile regression. The command estimates a semiparametric sample selection model without instrument or large support regressor, and outputs the point estimates of the homogenous linear coefficients, their bootstrap standard errors, as well as the p-value for a specification test.

In this paper, we build a new test of rational expectations based on the marginal distributions of realizations and subjective beliefs. This test is widely applicable, including in the common situation where realizations and beliefs are observed in two different datasets that cannot be matched. We show that whether one can rationalize rational expectations is equivalent to the distribution of realizations being a mean-preserving spread of the distribution of beliefs. The null hypothesis can then be rewritten as a system of many moment inequality and equality constraints, for which tests have been recently developed in the literature. Next, we go beyond testing by defining and estimating the minimal deviations from rational expectations that can be rationalized by the data. In the context of structural models, we build on this concept to propose an easy-to-implement way to conduct a sensitivity analysis on the assumed form of expectations. Finally, we apply our framework to test for and quantify deviations from rational expectations about future earnings, and examine the consequences of such departures in the context of a life-cycle model of consumption.

We show that data on subjective expectations, especially on outcomes from counterfactual choices and choice probabilities, are a powerful tool in recovering ex ante treatment effects as well as preferences for different treatments. In this paper we focus on the choice of occupation, and use elicited beliefs from a sample of male undergraduates at Duke University. By asking individuals about potential earnings associated with counterfactual choices of college majors and occupations, we can recover the distribution of the ex ante monetary returns to particular occupations, and how these returns vary across majors. We then propose a model of occupational choice which allows us to link subjective data on earnings and choice probabilities with the non-pecuniary preferences for each occupation. We find large differences in expected earnings across occupations, and substantial heterogeneity across individuals in the corresponding ex ante returns. However, while sorting across occupations is partly driven by the ex ante monetary returns, non-monetary factors play a key role in this decision. Finally, our results point to the existence of sizable complementarities between college major and occupations, both in terms of earnings and non-monetary benefits.

We consider the estimation of a semiparametric location-scale model subject to endogenous selection, in the absence of an instrument or a large support regressor. Identification relies on the independence between the covariates and selection, for arbitrarily large values of the outcome. In this context, we propose a simple estimator, which combines extremal quantile regressions with minimum distance. We establish the asymptotic normality of this estimator by extending previous results on extremal quantile regressions to allow for selection. Finally, we apply our method to estimate the black-white wage gap among males from the NLSY79 and NLSY97. We find that premarket factors such as AFQT and family background characteristics play a key role in explaining the level and evolution of the black-white wage gap.