Program Evaluation and Causal Inference With High‐Dimensional Data

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) and local quantile treatment effects (LQTE) in data‐rich environments. We can handle very many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function‐valued outcomes. Our framework covers the special case of exogenous receipt of treatment, either conditional on controls or unconditionally as in randomized control trials. In the latter case, our approach produces efficient estimators and honest bands for (functional) average treatment effects (ATE) and quantile treatment effects (QTE). To make informative inference possible, we assume that key reduced‐form predictive relationships are approximately sparse. This assumption allows the use of regularization and selection methods to estimate those relations, and we provide methods for post‐regularization and post‐selection inference that are uniformly valid (honest) across a wide range of models. We show that a key ingredient enabling honest inference is the use of orthogonal or doubly robust moment conditions in estimating certain reduced‐form functional parameters. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) eligibility and participation on accumulated assets. The results on program evaluation are obtained as a consequence of more general results on honest inference in a general moment‐condition framework, which arises from structural equation models in econometrics. Here, too, the crucial ingredient is the use of orthogonal moment conditions, which can be constructed from the initial moment conditions. We provide results on honest inference for (function‐valued) parameters within this general framework where any high‐quality, machine learning methods (e.g., boosted trees, deep neural networks, random forest, and their aggregated and hybrid versions) can be used to learn the nonparametric/high‐dimensional components of the model. These include a number of supporting auxiliary results that are of major independent interest: namely, we (1) prove uniform validity of a multiplier bootstrap, (2) offer a uniformly valid functional delta method, and (3) provide results for sparsity‐based estimation of regression functions for function‐valued outcomes.     

Robust Confidence Regions for Incomplete Models

Call an economic model incomplete if it does not generate a probabilistic prediction even given knowledge of all parameter values. We propose a method of inference about unknown parameters for such models that is robust to heterogeneity and dependence of unknown form. The key is a Central Limit Theorem for belief functions; robust confidence regions are then constructed in a fashion paralleling the classical approach. Monte Carlo simulations support tractability of the method and demonstrate its enhanced robustness relative to existing methods.     

Grouped Patterns of Heterogeneity in Panel Data

This paper introduces time‐varying grouped patterns of heterogeneity in linear panel data models. A distinctive feature of our approach is that group membership is left unrestricted. We estimate the parameters of the model using a “grouped fixed‐effects” estimator that minimizes a least squares criterion with respect to all possible groupings of the cross‐sectional units. Recent advances in the clustering literature allow for fast and efficient computation. We provide conditions under which our estimator is consistent as both dimensions of the panel tend to infinity, and we develop inference methods. Finally, we allow for grouped patterns of unobserved heterogeneity in the study of the link between income and democracy across countries.     

Sign Restrictions,Structural Vector Autoregressions,and Useful Prior Information

This paper makes the following original contributions to the literature. (i) We develop a simpler analytical characterization and numerical algorithm for Bayesian inference in structural vector autoregressions (VARs) that can be used for models that are overidentified, just‐identified, or underidentified. (ii) We analyze the asymptotic properties of Bayesian inference and show that in the underidentified case, the asymptotic posterior distribution of contemporaneous coefficients in an n‐variable VAR is confined to the set of values that orthogonalize the population variance–covariance matrix of ordinary least squares residuals, with the height of the posterior proportional to the height of the prior at any point within that set. For example, in a bivariate VAR for supply and demand identified solely by sign restrictions, if the population correlation between the VAR residuals is positive, then even if one has available an infinite sample of data, any inference about the demand elasticity is coming exclusively from the prior distribution. (iii) We provide analytical characterizations of the informative prior distributions for impulse‐response functions that are implicit in the traditional sign‐restriction approach to VARs, and we note, as a special case of result (ii), that the influence of these priors does not vanish asymptotically. (iv) We illustrate how Bayesian inference with informative priors can be both a strict generalization and an unambiguous improvement over frequentist inference in just‐identified models. (v) We propose that researchers need to explicitly acknowledge and defend the role of prior beliefs in influencing structural conclusions and we illustrate how this could be done using a simple model of the U.S. labor market.     

A Practical Two‐Step Method for Testing Moment Inequalities

This paper considers the problem of testing a finite number of moment inequalities. We propose a two‐step approach. In the first step, a confidence region for the moments is constructed. In the second step, this set is used to provide information about which moments are “negative.” A Bonferonni‐type correction is used to account for the fact that, with some probability, the moments may not lie in the confidence region. It is shown that the test controls size uniformly over a large class of distributions for the observed data. An important feature of the proposal is that it remains computationally feasible, even when the number of moments is large. The finite‐sample properties of the procedure are examined via a simulation study, which demonstrates, among other things, that the proposal remains competitive with existing procedures while being computationally more attractive.     

Quantile Selection Models With an Application to Understanding Changes in Wage Inequality

We propose a method to correct for sample selection in quantile regression models. Selection is modeled via the cumulative distribution function, or copula, of the percentile error in the outcome equation and the error in the participation decision. Copula parameters are estimated by minimizing a method‐of‐moments criterion. Given these parameter estimates, the percentile levels of the outcome are readjusted to correct for selection, and quantile parameters are estimated by minimizing a rotated “check” function. We apply the method to correct wage percentiles for selection into employment, using data for the UK for the period 1978–2000. We also extend the method to account for the presence of equilibrium effects when performing counterfactual exercises.     

Nearly Optimal Tests When a Nuisance Parameter Is Present Under the Null Hypothesis

This paper considers nonstandard hypothesis testing problems that involve a nuisance parameter. We establish an upper bound on the weighted average power of all valid tests, and develop a numerical algorithm that determines a feasible test with power close to the bound. The approach is illustrated in six applications: inference about a linear regression coefficient when the sign of a control coefficient is known; small sample inference about the difference in means from two independent Gaussian samples from populations with potentially different variances; inference about the break date in structural break models with moderate break magnitude; predictability tests when the regressor is highly persistent; inference about an interval identified parameter; and inference about a linear regression coefficient when the necessity of a control is in doubt.     

Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data‐dependent convex combination of two identification‐robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi‐conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug‐in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification‐robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.     

Moment Inequalities and Their Application

This paper provides conditions under which the inequality constraints generated by either single agent optimizing behavior or the best response condition of multiple agent problems can be used as a basis for estimation and inference. An application illustrates how the use of these inequality constraints can simplify the analysis of complex behavioral models.     

Spurious Inference in Reduced‐Rank Asset‐Pricing Models

This note studies some seemingly anomalous results that arise in possibly misspecified, reduced‐rank linear asset‐pricing models estimated by the continuously updated generalized method of moments. When a spurious factor (that is, a factor that is uncorrelated with the returns on the test assets) is present, the test for correct model specification has asymptotic power that is equal to the nominal size. In other words, applied researchers will erroneously conclude that the model is correctly specified even when the degree of misspecification is arbitrarily large. The rejection probability of the test for overidentifying restrictions typically decreases further in underidentified models where the dimension of the null space is larger than 1.     

Bootstrap Testing of Hypotheses on Co‐Integration Relations in Vector Autoregressive Models

It is well known that the finite‐sample properties of tests of hypotheses on the co‐integrating vectors in vector autoregressive models can be quite poor, and that current solutions based on Bartlett‐type corrections or bootstrap based on unrestricted parameter estimators are unsatisfactory, in particular in those cases where also asymptotic χ² tests fail most severely. In this paper, we solve this inference problem by showing the novel result that a bootstrap test where the null hypothesis is imposed on the bootstrap sample is asymptotically valid. That is, not only does it have asymptotically correct size, but, in contrast to what is claimed in existing literature, it is consistent under the alternative. Compared to the theory for bootstrap tests on the co‐integration rank (Cavaliere, Rahbek, and Taylor, 2012), establishing the validity of the bootstrap in the framework of hypotheses on the co‐integrating vectors requires new theoretical developments, including the introduction of multivariate Ornstein–Uhlenbeck processes with random (reduced rank) drift parameters. Finally, as documented by Monte Carlo simulations, the bootstrap test outperforms existing methods.     

Inferring Labor Income Risk and Partial Insurance From Economic Choices

This paper uses the information contained in the joint dynamics of individuals' labor earnings and consumption‐choice decisions to quantify both the amount of income risk that individuals face and the extent to which they have access to informal insurance against this risk. We accomplish this task by using indirect inference to estimate a structural consumption–savings model, in which individuals both learn about the nature of their income process and partly insure shocks via informal mechanisms. In this framework, we estimate (i) the degree of partial insurance, (ii) the extent of systematic differences in income growth rates, (iii) the precision with which individuals know their own income growth rates when they begin their working lives, (iv) the persistence of typical labor income shocks, (v) the tightness of borrowing constraints, and (vi) the amount of measurement error in the data. In implementing indirect inference, we find that an auxiliary model that approximates the true structural equations of the model (which are not estimable) works very well, with negligible small sample bias. The main substantive findings are that income shocks are moderately persistent, systematic differences in income growth rates are large, individuals have substantial amounts of information about their income growth rates, and about one‐half of income shocks are smoothed via partial insurance. Putting these findings together, the amount of uninsurable lifetime income risk that individuals perceive is substantially smaller than what is typically assumed in calibrated macroeconomic models with incomplete markets.     

Poor (Wo)man's Bootstrap

The bootstrap is a convenient tool for calculating standard errors of the parameter estimates of complicated econometric models. Unfortunately, the fact that these models are complicated often makes the bootstrap extremely slow or even practically infeasible. This paper proposes an alternative to the bootstrap that relies only on the estimation of one‐dimensional parameters. We introduce the idea in the context of M and GMM estimators. A modification of the approach can be used to estimate the variance of two‐step estimators.     

Sieve Wald and QLR Inferences on Semi/Nonparametric Conditional Moment Models

This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. These models are often ill‐posed and hence it is difficult to verify whether a (possibly nonlinear) functional is root‐n estimable or not. We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is root‐n estimable or not. We establish the following new useful results: (1) the asymptotic normality of a plug‐in penalized sieve minimum distance (PSMD) estimator of a (possibly nonlinear) functional; (2) the consistency of simple sieve variance estimators for the plug‐in PSMD estimator, and hence the asymptotic chi‐square distribution of the sieve Wald statistic; (3) the asymptotic chi‐square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non‐optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) asymptotic properties of sieve Wald and SQLR for functionals of increasing dimension. Simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.     

Robust Confidence Intervals for Average Treatment Effects Under Limited Overlap

Limited overlap between the covariate distributions of groups with different treatment assignments does not only make estimates of average treatment effects rather imprecise, but can also lead to substantially distorted confidence intervals. This paper argues that this is because the coverage error of traditional confidence intervals is driven by the number of observations in the areas of limited overlap. Some of these “local sample sizes” can be very small in applications, up to the point that distributional approximations derived from classical asymptotic theory become unreliable. Building on this observation, this paper constructs confidence intervals based on classical approaches to small sample inference. The approach is easy to implement, and has superior theoretical and practical properties relative to standard methods in empirically relevant settings.     

Parametric Inference and Dynamic State Recovery From Option Panels

We develop a new parametric estimation procedure for option panels observed with error. We exploit asymptotic approximations assuming an ever increasing set of option prices in the moneyness (cross‐sectional) dimension, but with a fixed time span. We develop consistent estimators for the parameters and the dynamic realization of the state vector governing the option price dynamics. The estimators converge stably to a mixed‐Gaussian law and we develop feasible estimators for the limiting variance. We also provide semiparametric tests for the option price dynamics based on the distance between the spot volatility extracted from the options and one constructed nonparametrically from high‐frequency data on the underlying asset. Furthermore, we develop new tests for the day‐by‐day model fit over specific regions of the volatility surface and for the stability of the risk‐neutral dynamics over time. A comprehensive Monte Carlo study indicates that the inference procedures work well in empirically realistic settings. In an empirical application to S&P 500 index options, guided by the new diagnostic tests, we extend existing asset pricing models by allowing for a flexible dynamic relation between volatility and priced jump tail risk. Importantly, we document that the priced jump tail risk typically responds in a more pronounced and persistent manner than volatility to large negative market shocks.     

Fixed‐Smoothing Asymptotics in a Two‐Step Generalized Method of Moments Framework

This paper develops the fixed‐smoothing asymptotics in a two‐step generalized method of moments (GMM) framework. Under this type of asymptotics, the weighting matrix in the second‐step GMM criterion function converges weakly to a random matrix and the two‐step GMM estimator is asymptotically mixed normal. Nevertheless, the Wald statistic, the GMM criterion function statistic, and the Lagrange multiplier statistic remain asymptotically pivotal. It is shown that critical values from the fixed‐smoothing asymptotic distribution are high order correct under the conventional increasing‐smoothing asymptotics. When an orthonormal series covariance estimator is used, the critical values can be approximated very well by the quantiles of a noncentral F distribution. A simulation study shows that statistical tests based on the new fixed‐smoothing approximation are much more accurate in size than existing tests.     

Generalized Instrumental Variable Models

This paper develops characterizations of identified sets of structures and structural features for complete and incomplete models involving continuous or discrete variables. Multiple values of unobserved variables can be associated with particular combinations of observed variables. This can arise when there are multiple sources of heterogeneity, censored or discrete endogenous variables, or inequality restrictions on functions of observed and unobserved variables. The models generalize the class of incomplete instrumental variable (IV) models in which unobserved variables are single‐valued functions of observed variables. Thus the models are referred to as generalized IV (GIV) models, but there are important cases in which instrumental variable restrictions play no significant role. Building on a definition of observational equivalence for incomplete models the development uses results from random set theory that guarantee that the characterizations deliver sharp bounds, thereby dispensing with the need for case‐by‐case proofs of sharpness. The use of random sets defined on the space of unobserved variables allows identification analysis under mean and quantile independence restrictions on the distributions of unobserved variables conditional on exogenous variables as well as under a full independence restriction. The results are used to develop sharp bounds on the distribution of valuations in an incomplete model of English auctions, improving on the pointwise bounds available until now. Application of many of the results of the paper requires no familiarity with random set theory.     

Power Enhancement in High‐Dimensional Cross‐Sectional Tests

We propose a novel technique to boost the power of testing a high‐dimensional vector H : θ = 0 against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating high‐dimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a “power enhancement component,” which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. The null distribution does not require stringent regularity conditions, and is completely determined by that of the pivotal statistic. The proposed methods are then applied to testing the factor pricing models and validating the cross‐sectional independence in panel data models.     

A Structural Model of Dense Network Formation

This paper proposes an empirical model of network formation, combining strategic and random networks features. Payoffs depend on direct links, but also link externalities. Players meet sequentially at random, myopically updating their links. Under mild assumptions, the network formation process is a potential game and converges to an exponential random graph model (ERGM), generating directed dense networks. I provide new identification results for ERGMs in large networks: if link externalities are nonnegative, the ERGM is asymptotically indistinguishable from an Erdős–Rényi model with independent links. We can identify the parameters only when at least one of the externalities is negative and sufficiently large. However, the standard estimation methods for ERGMs can have exponentially slow convergence, even when the model has asymptotically independent links. I thus estimate parameters using a Bayesian MCMC method. When the parameters are identifiable, I show evidence that the estimation algorithm converges in almost quadratic time.