Andres Santos

Inference in Nonparametric Instrumental Variables With Partial Identification

This paper develops methods for hypothesis testing in a nonparametric instrumental variables setting within a partial identification framework. We construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction. The same test statistic can be employed under identification, in which case the hypothesis is whether the true model satisfies the posited property. An almost sure consistent bootstrap procedure is provided for obtaining critical values. Possible applications include testing for semiparametric specifications as well as building confidence regions for certain functionals on the identified set. As an illustration we obtain confidence intervals for the level and slope of Brazilian fuel Engel curves. A Monte Carlo study examines finite sample performance.

On the testability of identification in some nonparametric models with endogeneity

This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.

Sensitivity to missing data assumptions: Theory and an evaluation of the U.S. wage structure

This paper develops methods for assessing the sensitivity of empirical conclusions regarding conditional distributions to departures from the missing at random (MAR) assumption. We index the degree of non-ignorable selection governing the missingness process by the maximal Kolmogorov-Smirnov (KS) distance between the distributions of missing and observed outcomes across all values of the covariates. Sharp bounds on minimum mean square approximations to conditional quantiles are derived as a function of the nominal level of selection considered in the sensitivity analysis and a weighted bootstrap procedure is developed for conducting inference. Using these techniques, we conduct an empirical assessment of the sensitivity of observed earnings patterns in U.S. Census data to deviations from the MAR assumption. We find that the well-documented increase in the returns to schooling between 1980 and 1990 is relatively robust to deviations from the missing at random assumption except at the lowest quantiles of the distribution, but that conclusions regarding heterogeneity in returns and changes in the returns function between 1990 and 2000 are very sensitive to departures from ignorability.

Asymptotically Efficient Estimation of Models Defined by Convex Moment Inequalities

This paper examines the efficient estimation of partially identified models defined by moment inequalities that are convex in the parameter of interest. In such a setting, the identified set is itself convex and hence fully characterized by its support function. We provide conditions under which, despite being an infinite dimensional parameter, the support function admits √n‐consistent regular estimators. A semiparametric efficiency bound is then derived for its estimation, and it is shown that any regular estimator attaining it must also minimize a wide class of asymptotic loss functions. In addition, we show that the “plug‐in” estimator is efficient, and devise a consistent bootstrap procedure for estimating its limiting distribution. The setting we examine is related to an incomplete linear model studied in Beresteanu and Molinari (2008) and Bontemps, Magnac, and Maurin (2012), which further enables us to establish the semiparametric efficiency of their proposed estimators for that problem.

Semi‐Parametric Estimation of Non‐Separable Models: A Minimum Distance from Independence Approach

This paper focuses on nonseparable structural models of the form Y = m(X, U, α0) with U X and in which the structural parameter α0 contains both finite dimensional (θ0) and infinite dimensional (h0) unknown components. Our proposal is to estimate α0 by a minimum distance from independence (MDI) criterion. We show that: (i) our estimator of h0 is consistent and obtain rates of convergence; (ii) the estimator of θ0 is square root n consistent and asymptotically normally distributed.

The wild bootstrap with a "small" number of "large" clusters -

This paper studies the properties of the wild bootstrap-based test proposed in Cameron et al. (2008) in settings with clustered data. Cameron et al. (2008) provide simulations that suggest this test works well even in settings with as few as five clusters, but existing theoretical analyses of its properties all rely on an asymptotic framework in which the number of clusters is “large.” In contrast to these analyses, we employ an asymptotic framework in which the number of clusters is “small,” but the number of observations per cluster is “large.” In this framework, we provide conditions under which the limiting rejection probability of an un-Studentized version of the test does not exceed the nominal level. Importantly, these conditions require, among other things, certain homogeneity restrictions on the distribution of covariates. We further establish that the limiting rejection probability of a Studentized version of the test does not exceed the nominal level by more than an amount that decreases exponentially with the number of clusters. We study the relevance of our theoretical results for finite samples via a simulation study.

Overidentification in Regular Models

In models defined by unconditional moment restrictions, specification tests are possible and estimators can be ranked in terms of efficiency whenever the number of moment restrictions exceeds the number of parameters. We show that a similar relationship between potential refutability of a model and semiparametric efficiency is present in a much broader class of settings. Formally, we show a condition we name local overidentification is required for both specification tests to have power against local alternatives and for the existence of both efficient and inefficient estimators of regular parameters. Our results immediately imply semiparametric conditional moment restriction models are typically locally overidentified, and hence their proper specification is locally testable. We further study nonparametric conditional moment restriction models and obtain a simple characterization of local overidentification in that context. As a result, we are able to determine when nonparametric conditional moment restriction models are locally testable, and when plug-in and two stage estimators of regular parameters are semiparametrically efficient.