ANDREWS, GENTZKOW, AND SHAPIRO (2020), henceforth AGS, make important progress on a challenging and fundamental question. The authors consider a setting in which a researcher reports an estimate ĉn that is unbiased under the researcher’s model. The researcher also reports a statistic γ̂n and a measure of informativeness of γ̂n for ĉn, denoted by . AGS show can be related to properties of the distribution of ĉn under models larger than the one employed by the researcher. As a result, can play a role in reassuring a reader who is concerned about some aspects of the researcher’s model. I illustrate the properties of with two examples. The first example is based on the classic selection problem of Heckman (1979) and portrays a setting in which a high is not reassuring. The example shows that if a researcher has assumed random assignment and the reader is concerned about it, then the bias of ĉn under the reader’s model can be arbitrarily large regardless of . The second example is based on the discussion of Attanasio, Meghir, and Santiago (2012) in AGS. This example illustrates that if ĉn and γ̂n are estimators of a common parameter, then can be linked to the properties of a Hausman (1978) test. As a result, √ 1 − can be shown to equal the (normalized) largest bias of ĉn under the reader’s model and, in this sense, a high is reassuring. I show this conclusion holds generally in a classical local asymptotic framework (Bickel, Klassen, Ritov, and Wellner (1993)). The two examples underscore that, as emphasized by AGS, whether a high is reassuring to the reader can crucially depend on the particulars of the researcher’s and reader’s models. I employ the notation in AGS with some minor modifications. The researcher’s model is indexed by η ∈ H. The reader entertains a larger model indexed by (η ζ) ∈ H × Z. I focus on the perspective of the reader, who views the parameter of interest as a function of (η ζ) that we denote by c(η ζ). Under the reader’s model, the distribution of the data is determined by the value of (η ζ). To emphasize this dependence, I write E(η ζ)[ĉn] to denote the expectation of ĉn under (η ζ). The property that ĉn is unbiased under the researcher’s model is equivalent to E(η 0)[ĉn] = c(η 0) for any η ∈H, while the bias under (η ζ) in the reader’s model is given by E(η ζ)[ĉn] − c(η ζ). As in AGS, I consider an analysis local to proper specification, meaning we study (η ζ) that are close to some (η0 0). Since E(η0 0)[ĉn] = c(η0 0) due to ĉn being unbiased under the researcher’s model, the bias of ĉn under any (η ζ) in the reader’s model equals E(η ζ)[ĉn] − c(η ζ)

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.

In the unconditional moment restriction model of Hansen (), specification tests and more efficient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest. We show that a similar relationship between potential refutability of a model and existence of more efficient estimators is present in much broader settings. Specifically, a condition we name local overidentification is shown to be equivalent to both the existence of specification tests with nontrivial local power and the existence of more efficient estimators of some “smooth” parameters in general semi/nonparametric models. Under our notion of local overidentification, various locally nontrivial specification tests such as Hausman tests, incremental Sargan tests (or optimally weighted quasi likelihood ratio tests) naturally extend to general semi/nonparametric settings. We further obtain simple characterizations of local overidentification for general models of nonparametric conditional moment restrictions with possibly different conditioning sets. The results are applied to determining when semi/nonparametric models with endogeneity are locally testable, and when nonparametric plug‐in and semiparametric two‐step GMM estimators are semiparametrically efficient. Examples of empirically relevant semi/nonparametric structural models are presented.

We study instrumental variables models characterized by: (i) Unobserved heterogeneity consisting of potential outcomes and response types that describe how the instrument determines treatment choice; (ii) Conditional independence of the instrument and the unobserved heterogeneity; and (iii) Convex restrictions on the distribution of unobserved heterogeneity. We show certain causal parameters are identified in these models if and only if a version of the kappa of Abadie (2003) exists. Our identification results are constructive in yielding estimating moment conditions. Focusing on a leading special case, we develop asymptotically normal estimators based on a doubly robust version of these moment conditions.

We show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the (exponential) rate at which the probability of a Type I error tends to zero unless the possible distributions for the observed data are restricted appropriately. From this, it follows that for the optimality claim for empirical likelihood in Kitamura (2001) to hold, additional assumptions and qualifications are required. Under stronger assumptions than those in Kitamura (2001), we establish the following optimality result: (i) empirical likelihood controls the rate at which the probability of a Type I error tends to zero and (ii) among all procedures for which the probability of a Type I error tends to zero at least as fast, empirical likelihood maximizes the rate at which the probability of a Type II error tends to zero for most alternatives. This result further implies that empirical likelihood maximizes the rate at which the probability of a Type II error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their Type I error probabilities.

This article studies an asymptotic framework for conducting inference on parameters of the form φ( heta_0), where φ is a known directionally differentiable function and heta_0 is estimated by $\hat heta_n$. In these settings, the asymptotic distribution of the plug-in estimator $φ(\hat heta_n)$ can be derived employing existing extensions to the Delta method. We show, however, that (full) differentiability of φ is a necessary and sufficient condition for bootstrap consistency whenever the limiting distribution of $\hat heta_n$ is Gaussian. An alternative resampling scheme is proposed that remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of φ. These results enable us to reduce potentially challenging statistical problems to simple analytical calculations—a feature we illustrate by developing a test of whether an identified parameter belongs to a convex set. We highlight the empirical relevance of our results by conducting inference on the qualitative features of trends in (residual) wage inequality in the U.S.

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.

Shape restrictions have played a central role in economics as both testable implications of theory and sufficient conditions for obtaining informative counterfactual predictions. In this paper, we provide a general procedure for inference under shape restrictions in identified and partially identified models defined by conditional moment restrictions. Our test statistics and proposed inference methods are based on the minimum of the generalized method of moments (GMM) objective function with and without shape restrictions. Uniformly valid critical values are obtained through a bootstrap procedure that approximates a subset of the true local parameter space. In an empirical analysis of the effect of childbearing on female labor supply, we show that employing shape restrictions in linear instrumental variables (IV) models can lead to shorter confidence regions for both local and average treatment effects. Other applications we discuss include inference for the variability of quantile IV treatment effects and for bounds on average equivalent variation in a demand model with general heterogeneity.

This paper studies the wild bootstrap–based test proposed in Cameron, Gelbach, and Miller (2008). Existing analyses of its properties require that number of clusters is “large.” In an asymptotic framework in which the number of clusters is “small,” we provide conditions under which an unstudentized version of the test is valid. These conditions include homogeneity-like restrictions on the distribution of covariates. We further establish that a studentized version of the test may only overreject the null hypothesis by a “small” amount that decreases exponentially with the number of clusters. We obtain a qualitatively similar result for “score” bootstrap-based tests, which permit testing in nonlinear models.

We propose a method for using instrumental variables (IV) to draw inference about causal effects for individuals other than those affected by the instrument at hand. Policy relevance and external validity turn on the ability to do this reliably. Our method exploits the insight that both the IV estimand and many treatment parameters can be expressed as weighted averages of the same underlying marginal treatment effects. Since the weights are identified, knowledge of the IV estimand generally places some restrictions on the unknown marginal treatment effects, and hence on the values of the treatment parameters of interest. We show how to extract information about the treatment parameter of interest from the IV estimand and, more generally, from a class of IV‐like estimands that includes the two stage least squares and ordinary least squares estimands, among others. Our method has several applications. First, it can be used to construct nonparametric bounds on the average causal effect of a hypothetical policy change. Second, our method allows the researcher to flexibly incorporate shape restrictions and parametric assumptions, thereby enabling extrapolation of the average effects for compliers to the average effects for different or larger populations. Third, our method can be used to test model specification and hypotheses about behavior, such as no selection bias and/or no selection on gain.