Christoph Rothe

Nonparametric Regression with Nonparametrically Generated Covariates

We analyze the properties of non- and semiparametric estimation procedures involving nonparametric regression with generated covariates. Such estimators appear in numerous econometric applications, including nonparametric estimation of simultaneous equation models, sample selection models, treatment effect models, and censored regression models, but so far there seems to be no unified theory to establish their statistical properties. Our paper provides such results, allowing to establish asymptotic properties like rates of consistency or asymptotic normality for a wide range of semi- and nonparametric estimators. We also show how to account for the presence of nonparametrically generated regressors when computing standard errors.

Semiparametric Estimation with Generated Covariates

In this paper, we study a general class of semiparametric optimization estimators of a vector-valued parameter. The criterion function depends on two types of infinite-dimensional nuisance parameters: a conditional expectation function that has been estimated nonparametrically using generated covariates, and another estimated function that is used to compute the generated covariates in the first place. We study the asymptotic properties of estimators in this class, which is a nonstandard problem due to the presence of generated covariates. We give conditions under which estimators are root-n consistent and asymptotically normal, and derive a general formula for the asymptotic variance.

Partial Distributional Policy Effects

In this paper, we propose a method to evaluate the effect of a counterfactual change in the unconditional distribution of a single covariate on the unconditional distribution of an outcome variable of interest. Both fixed and infinitesimal changes are considered. We show that such effects are point identified under general conditions if the covariate affected by the counterfactual change is continuously distributed, but are typically only partially identified if its distribution is discrete. For the latter case, we derive informative bounds making use of the available information. We also discuss estimation and inference.

Misspecification Testing in a Class of Conditional Distributional Models

We propose a specification test for a wide range of parametric models for the conditional distribution function of an outcome variable given a vector of covariates. The test is based on the Cramer–von Mises distance between an unrestricted estimate of the joint distribution function of the data and a restricted estimate that imposes the structure implied by the model. The procedure is straightforward to implement, is consistent against fixed alternatives, has nontrivial power against local deviations of order n − 1/2 from the null hypothesis, and does not require the choice of smoothing parameters. In an empirical application, we use our test to study the validity of various models for the conditional distribution of wages in the United States.

Generated Covariates in Nonparametric Estimation: A Short Review

In many applications, covariates are not observed but have to be estimated from data. We outline some regression-type models where such a situation occurs and discuss estimation of the regression function in this context. We review theoretical results on how asymptotic properties of nonparametric estimators differ in the presence of generated covariates from the standard case where all covariates are observed. These results also extend to settings where the focus of interest is on average functionals of the regression function.

Decomposing the Composition Effect: The Role of Covariates in Determining Between-Group Differences in Economic Outcomes

In this article, we study the structure of the composition effect, which is the part of the observed between-group difference in the distribution of some economic outcome that can be explained by differences in the distribution of covariates. Using results from copula theory, we derive a new representation that contains three types of components: (i) the “direct contribution” of each covariate due to between-group differences in the respective marginal distributions, (ii) several “two-way” and “higher-order interaction effects” due to the interplay between two or more marginal distributions, and (iii) a “dependence effect” accounting for between-group differences in dependence patterns among the covariates. We show how these components can be estimated in practice, and use our method to study the evolution of the wage distribution in the United States between 1985 and 2005. We obtain some new and interesting empirical findings. For example, our estimates suggest that the dependence effect alone can explain about one-fifth of the increase in wage inequality over that period (as measured by the difference between the 90% and the 10% quantile).

Robust Confidence Intervals for Average Treatment Effects Under Limited Overlap

Estimators of average treatment effects under unconfounded treatment assignment are known to become rather imprecise if there is limited overlap in the covariate distributions between the treatment groups. But such limited overlap can also have a detrimental effect on inference, and lead for example to highly distorted confidence intervals. This paper shows that this is because the coverage error of traditional confidence intervals is not so much driven by the total sample size, but by the number of observations in the areas of limited overlap. At least some of these “local sample sizes” are often very small in applications, up to the point where distributional approximation derived from the Central Limit Theorem become unreliable. Building on this observation, the paper proposes two new robust confidence intervals that are extensions of classical approaches to small sample inference. It shows that these approaches are easy to implement, and have superior theoretical and practical properties relative to standard methods in empirically relevant settings. They should thus be useful for practitioners.