Methods

In nonlinear panel models with fixed effects and fixed-T, the incidental parameter problem poses identification difficulties for structural parameters and partial effects. Existing solutions are model-specific, likelihood-based, impose time homogeneity, or restrict the distribution of unobserved heterogeneity. We provide new identification results for the large class of Fixed Effects Linear Transformation (FELT) models with unknown, time-varying, weakly monotone transformation functions.

Robins et al. (2008, 2016b) applied the theory of higher order infuence functions (HOIFs) to derive an estimator of the mean of an outcome Y in a missing data model with Y missing at random conditional on a vector X of continuous covariates; their estimator, in contrast to previous estimators, is semiparametric efficient under minimal conditions. However the Robins et al. (2008, 2016b) estimator depends on a non-parametric estimate of the density of X. In this paper, we introduce a new HOIF estimator that has the same asymptotic properties as their estimator but does not require non-parametric estimation of a multivariate density, which is important because accurate estimation of a high dimensional density is not feasible at the moderate sample sizes often encountered in applications. We also show that our estimator can be generalized to the entire class of functionals considered by Robins et al. (2008) which include the average effect of a treatment on a response Y when a vector X suffices to control confounding and the expected conditional variance of a response Y given a vector X.

If the productive characteristics of workers and firms are truly multi-dimensional, what features of the data do we miss by modeling them as one-dimensional scalars? This is the question we ask in this paper.

The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functions contained within.

We revisit the classic semiparametric problem of inference on a low dimensional parameter θ0 in the presence of high-dimensional nuisance parameters η0.

This paper studies identification and estimation of the distribution of bidder valuations in an incomplete model of English auctions. As in Haile and Tamer (2003) bidders are assumed to (i) bid no more than their valuations and (ii) never let an opponent win at a price they are willing to beat. Unlike the model studied by Haile and Tamer (2003), the requirement of independent private values is dropped, enabling the use of these restrictions on bidder behavior with affiliated private values, for example through the presence of auction specifi…c unobservable heterogeneity. In addition, a semiparametric index restriction on the effect of auction-specifi…c observable heterogeneity is incorporated, which, relative to nonparametric methods, can be help- ful in alleviating the curse of dimensionality with a moderate or large number of covariates. The identification analysis employs results from Chesher and Rosen (2017) to characterize identified sets for bidder valuation distributions and functionals thereof.

This paper studies inference on fixed effects in a linear regression model estimated from network data. An important special case of our setup is the two-way regression model, which is a workhorse method in the analysis of matched data sets. Networks are typically quite sparse and it is difficult to see how the data carry information about certain parameters. We derive bounds on the variance of the fixed-effect estimator that uncover the importance of the structure of the network. These bounds depend on the smallest non-zero eigenvalue of the (normalized) Laplacian of the network and on the degree structure of the network. The Laplacian is a matrix that describes the network and its smallest non-zero eigenvalue is a measure of connectivity, with smaller values indicating less-connected networks. These bounds yield conditions for consistent estimation and convergence rates, and allow to evaluate the accuracy of first-order approximations to the variance of the fixed-effect estimator. The bounds are also used to assess the bias and variance of estimators of moments of the fixed effects.

This paper studies inference for the average treatment effect in randomized controlled trials with covariate-adaptive randomization. Here, by covariate-adaptive randomization, we mean randomization schemes that first stratify according to baseline covariates and then assign treatment status so as to achieve "balance" within each stratum.

Doubly Robust Uniform Confidence Band for the Conditional Average Treatment Effect Function

One of the main objectives of empirical analysis of experiments and quasi-experiments is to inform policy decisions that determine the allocation of treatments to individuals with different observable covariates. We study the properties and implementation of the Empirical Welfare Maximization (EWM) method, which estimates a treatment assignment policy by maximizing the sample analog of average social welfare over a class of candidate treatment policies. The EWM approach is attractive in terms of both statistical performance and practical implementation in realistic settings of policy design. Common features of these settings include: (i) feasible treatment assignment rules are constrained exogenously for ethical, legislative, or political reasons, (ii) a policy maker wants a simple treatment assignment rule based on one or more eligibility scores in order to reduce the dimensionality of individual observable characteristics, and/or (iii) the proportion of individuals who can receive the treatment is a priori limited due to a budget or a capacity constraint. We show that when the propensity score is known, the average social welfare attained by EWM rules converges at least at n-1=2 rate to the maximum obtainable welfare uniformly over a minimally constrained class of data distributions, and this uniform convergence rate is minimax optimal. We examine how the uniform convergence rate depends on the richness of the class of candidate decision rules, the distribution of conditional treatment effects, and the lack of knowledge of the propensity score. We other easily implementable algorithms for computing the EWM rule and an application using experimental data from the National JTPA Study.

This paper provides a method to construct simultaneous confidence bands for quantile and quantile effect functions for possibly discrete or mixed discrete-continuous random variables. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparamet-ric, and nonparametric modeling strategies and does not depend on the sampling schemes. It is based upon projection of simultaneous confidence bands for distribution functions. We apply our method to analyze the distributional impact of insurance coverage on health care utilization and to provide a distributional decomposition of the racial test score gap. Our analysis generates new interesting findings, and com-plements previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering standard inference methods invalid for obtaining uniform confidence bands for quantile and quantile effects functions.

We study high-dimensional linear models with error-in-variables. Such models are motivated by various applications in econometrics, finance and genetics. These models are challenging because of the need to account for measurement errors to avoid non-vanishing biases in addition to handle the high dimensionality of the parameters. A recent growing literature has proposed various estimators that achieve good rates of convergence. Our main contribution complements this literature with the construction of simultaneous confidence regions for the parameters of interest in such high-dimensional linear models with error-in-variables. These confidence regions are based on the construction of moment conditions that have an additional orthogonality property with respect to nuisance parameters. We provide a construction that requires us to estimate an auxiliary high-dimensional linear model with error-in-variables for each component of interest. We use a multiplier bootstrap to compute critical values for simultaneous confidence intervals for a target subset of the components. We show its validity despite of possible (moderate) model selection mistakes, and allowing the number of target coefficients to be larger than the sample size. We apply and discuss the implications of our results to two examples and conduct Monte Carlo simulations to illustrate the performance of the proposed procedure for each variable whose coefficient is the target of inference.

In the regression discontinuity design (RDD), it is common practice to asses the credibility of the design by testing whether the means of baseline covariates do not change at the cutoff(or threshold) of the running variable. This practice is partly motivated by the stronger im-plication derived by Lee (2008), who showed that under certain conditions the distribution of baseline covariates in the RDD must be continuous at the cutoff. We propose a permutation test based on the so-called induced ordered statistics for the null hypothesis of continuity of the distribution of baseline covariates at the cutoff; and introduce a novel asymptotic framework to analyze its properties. The asymptotic framework is intended to approximate a small sample phenomenon: even though the total number n of observations may be large, the number of effective observations local to the cutoff is often small. Thus, while traditional asymptotics in RDD require a growing number of observations local to the cutoff as n → ∞, our framework keeps the number q of observations local to the cutoff fixed as n → ∞. The new test is easy to implement, asymptotically valid under weak conditions, exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity, and has favorable power properties relative to tests based on means. In a simulation study, we find that the new test controls size remarkably well across designs. We then use our test to evaluate the plausibility of the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.

A breakdown frontier is the boundary between the set of assumptions which lead to a specific conclusion and those which do not. In a potential outcomes model with a binary treatment, we consider two conclusions: First, that ATE is at least a specific value (e.g., nonnegative) and second that the proportion of units who benefit from treatment is at least a specific value (e.g., at least 50%). For these conclusions, we derive the breakdown frontier for two kinds of assumptions: one which indexes deviations from random assignment of treatment, and one which indexes deviations from rank invariance. These classes of assumptions nest both the point identifying assumptions of random assignment and rank invariance and the opposite end of no constraints on treatment selection or the dependence structure between potential outcomes. This frontier provides a quantitative measure of robustness of conclusions to deviations in the point identifying assumptions. We derive √N-consistent sample analog estimators for these frontiers. We then provide an asymptotically valid bootstrap procedure for constructing lower uniform confidence bands for the breakdown frontier. As a measure of robustness, this confidence band can be presented alongside traditional point estimates and confidence intervals obtained under point identifying assumptions. We illustrate this approach in an empirical application to the effect of child soldiering on wages. We find that conclusions are fairly robust to failure of rank invariance, when random assignment holds, but conclusions are much more sensitive to both assumptions for small deviations from random assignment.

The impact of measurement error in explanatory variables on quantile regression functions is investigated using a small variance approximation. The approximation shows how the error contaminated and error free quantile regression functions are related. A key factor is the distribution of the error free explanatory variable. Exact calculations probe the accuracy of the approximation. The order of the approximation error is unchanged if the density of the error free explanatory variable is replaced by the density of the error contaminated explanatory variable which is easily estimated. It is then possible to use the approximation to investigate the sensitivity of estimates to varying amounts of measurement error.

We derive conditions under which a demand function with nonseparable unobserved heterogeneity in tastes can be estimated consistently by nonparametric quantile regression subject to the shape restriction from the Slutsky inequality.

We consider a two-period Bayesian trading game where in each period informed agents decide whether to buy an asset ("government debt") after observing an idiosyncratic signal about the prospects of default. While second-period buyers only need to forecast default, first-period buyers pass the asset to the new agents in the secondary market, and thus need to form beliefs about the price that will prevail at that stage. We provide conditions such that coarser information in the hands of second-period agents makes the price of debt more resilient to bad shocks not only in the last period, but in the first one as well. We use this model to study the consequences of issuing debt denominated in domestic vs. foreign currency: we interpret the former as subject to inflation risk and the latter as subject to default risk, with inflation driven by the information of a less-sophisticated group of agents endowed with less precise information, and default by the information of sophisticated bond traders. Our results can be used to account for the behavior of debt prices across countries following the 2008 nancial crisis, and also provide a theory of "original sin."

Generalized Instrumental Variable Models