Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings

This paper reports estimates of the effects of JTPA training programs on the distribution of earnings. The estimation uses a new instrumental variable (IV) method that measures program impacts on quantiles. The quantile treatment effects (QTE) estimator reduces to quantile regression when selection for treatment is exogenously determined. QTE can be computed as the solution to a convex linear programming problem, although this requires first‐step estimation of a nuisance function. We develop distribution theory for the case where the first step is estimated nonparametrically. For women, the empirical results show that the JTPA program had the largest proportional impact at low quantiles. Perhaps surprisingly, however, JTPA training raised the quantiles of earnings for men only in the upper half of the trainee earnings distribution.     

Identification in Nonseparable Models

Weak nonparametric restrictions are developed, sufficient to identify the values of derivatives of structural functions in which latent random variables are nonseparable. These derivatives can exhibit stochastic variation. In a microeconometric context this allows the impact of a policy intervention, as measured by the value of a structural derivative, to vary across people who are identical as measured by covariates. When the restrictions are satisfied quantiles of the distribution of a policy impact across people can be identified. The identification restrictions are local in the sense that they are specific to the values of the covariates and the specific quantiles of latent variables at which identification is sought. The conditions do not include the commonly required independence of latent variables and covariates. They include local versions of the classical rank and order conditions and local quantile insensitivity conditions. Values of structural derivatives are identified by functionals of quantile regression functions and can be estimated using the same functionals applied to estimated quantile regression functions.     

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.     

基于非线性分位数回归模型的多期VaR风险测度

多期VaR主要受到持有期及波动率两个变量的影响,并且其影响模式(线性或非线性)的确定对于准确地进行VaR风险测度至关重要。非线性分位数回归模型,能够克服线性分位数回归模型只能揭示多期VaR及其影响因素之间线性依赖关系的局限,从而提高多期VaR风险测度的准确性。结合波动模型与两个非线性分位数回归方法:QRNN和SVQR,给出了多期VaR风险测度的三类方案:波动模型法、QRNN+波动模型法、SVQR+波动模型法。选取3个股票价格指数作为研究对象,考虑了6种不同形式的波动模型,得到了18个多期VaR风险测度方法进行实证比较,结果表明:波动模型选择影响到多期VaR风险测度效果;SVQR+波动模型法略优于QRNN+波动模型法,并且两者显著优于波动模型法。     

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.     

Nonparametric Identification under Discrete Variation

This paper provides weak conditions under which there is nonparametric interval identification of local features of a structural function that depends on a discrete endogenous variable and is nonseparable in latent variates. The function delivers values of a discrete or continuous outcome and instruments may be discrete valued. Application of the analog principle leads to quantile regression based interval estimators of values and partial differences of structural functions. The results are used to investigate the nonparametric identifying power of the quarter‐of‐birth instruments used in Angrist and Krueger's 1991 study of the returns to schooling.     

Quantile Regression under Misspecification,with an Application to the U.S. Wage Structure

Quantile regression (QR) fits a linear model for conditional quantiles just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean‐squared error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR minimizes a weighted mean‐squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile regression concept, similar to the relationship between partial regression and OLS. We also present asymptotic theory for the QR process under misspecification of the conditional quantile function. The approximation properties of QR are illustrated using wage data from the U.S. census. These results point to major changes in inequality from 1990 to 2000.     

指令不均衡与股票收益关系研究——基于大规模数据分位数回归的实证

在指令不均衡与股票收益关系研究中,常常遇到两个困难：第一,不同市场环境下,前者对后者存在异质影响;第二,往往涉及大规模数据处理。为此,运用大规模数据分位数回归的方法,一方面揭示不同分位点处指令不均衡对股票收益的异质影响,细致刻画两者之间关系;另一方面适应大规模数据建模要求,得到更为可靠的结果。以上证A股和深证A股为研究对象,通过大规模数据分位数回归方法,得到了比均值回归更多有用信息。实证结果表明：第一,在高分位点处,滞后1期指令不均衡对股票收益具有正向影响且呈现上升趋势,而在低分位点却具有负向影响;第二,控制当期指令不均衡后,滞后期指令不均衡对股票收益具有负向影响,且随着分位点的增加呈现下降趋势。这些结果意味着,指令不均衡对股票收益具有一定的解释能力和预测能力。     

Quantile and Probability Curves Without Crossing

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem (Bassett and Koenker (1982)). The method consists in sorting or monotone rearranging the original estimated non‐monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples, establish a functional delta method for rearrangement‐related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural distribution and quantile functions using data on Vietnam veteran status and earnings.     

Inference on Counterfactual Distributions

Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article, we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios, we derive joint functional central limit theorems and bootstrap validity results for regression‐based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function‐valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no‐effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States. As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.     

Average and Quantile Effects in Nonseparable Panel Models

Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time‐homogeneity conditions that are like “time is randomly assigned” or “time is an instrument.” Partial‐identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed‐effects estimator is not a consistent estimator of the identified average effect, and a consistent estimator is given. A simple estimator of identified quantile treatment effects is given, providing a solution to the important problem of estimating quantile treatment effects from panel data. Bounds for overall effects in static and dynamic models are given. The dynamic bounds provide a partial‐identification solution to the important problem of estimating the effect of state dependence in the presence of unobserved heterogeneity. The impact of T, the number of time periods, is shown by deriving shrinkage rates for the identified set as T grows. We also consider semiparametric, discrete‐choice models and find that semiparametric panel bounds can be much tighter than nonparametric bounds. Computationally convenient methods for semiparametric models are presented. We propose a novel inference method that applies in panel data and other settings and show that it produces uniformly valid confidence regions in large samples. We give empirical illustrations.     

Immigration,Growth, and Unemployment: Panel VAR Evidence from OECD Countries

This paper examines empirically the interaction between immigration and host country economic conditions. We employ panel vector autoregression (VAR) techniques for a large annual data set on 22 OECD countries over the period 1987–2009. The VAR approach addresses the endogeneity problem by allowing for endogenous interactions between the variables in the system. Our results provide evidence of migration contribution to host economic prosperity (positive impact on GDP per capita and negative impact on aggregate unemployment, native‐ and foreign‐born unemployment rates). We also find that migration is influenced by host economic conditions (migration responds positively to host GDP per capita and negatively to host total unemployment rate).     

Inference on the Quantile Regression Process

Tests based on the quantile regression process can be formulated like the classical Kolmogorov–Smirnov and Cramér–von–Mises tests of goodness–of–fit employing the theory of Bessel processes as in Kiefer (1959). However, it is frequently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as “the Durbin problem” since it was posed in Durbin (1973), for parametric empirical processes. In this paper we consider an approach to the Durbin problem involving a martingale transformation of the parametric empirical process suggested by Khmaladze (1981) and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and location–scale shift models that underlie much of classical econometric inference. The methods are illustrated with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania experiments, conducted in 1988–89, were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells.     

Structural Nonparametric Cointegrating Regression

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill‐posed inverse problems. In functional cointegrating regressions where the regressor is an integrated or near‐integrated time series, it is shown here that inverse and ill‐posed inverse problems do not arise. Instead, simple nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving straightforward asymptotics that are useable in practical work. It is further shown that use of augmented regression, as is common in linear cointegration modeling to address endogeneity, does not lead to bias reduction in nonparametric regression, but there is an asymptotic gain in variance reduction. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series when there is a single integrated or near‐integrated regressor. The methods may be applied to a range of empirical models where functional estimation of cointegrating relations is required.     

Estimating the causal effect of measured endogenous variables: A tutorial on experimentally randomized instrumental variables

Omitted variables create endogeneity and thus bias the estimation of the causal effect of measured variables on outcomes. Such measured variables are ubiquitous and include perceptions, attitudes, emotions, behaviors, and choices. Even experimental studies are not immune to the endogeneity problem. I propose a solution to this challenge: Experimentally randomized instrumental variables (ERIVs), which can correct for endogeneity bias via instrumental variable estimation. Such ERIVs can be generated in laboratory or field settings. Using perceptions as an example of a measured variable, I examine 74 recent articles from two top-tier management journals. The estimation methods commonly used exposed estimates to potential endogeneity bias; yet, authors incorrectly interpreted the estimated coefficients as causal in all cases. Then I demonstrate the mechanics of the ERIV procedure using simulated data and show how researchers can apply this methodology in a real experimental context.     

Nonparametric Instrumental Variables Estimation of a Quantile Regression Model

We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression “error” conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill‐posed inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean‐square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples.     

IV Quantile Regression for Group‐Level Treatments,With an Application to the Distributional Effects of Trade

We present a methodology for estimating the distributional effects of an endogenous treatment that varies at the group level when there are group‐level unobservables, a quantile extension of Hausman and Taylor, 1981. Because of the presence of group‐level unobservables, standard quantile regression techniques are inconsistent in our setting even if the treatment is independent of unobservables. In contrast, our estimation technique is consistent as well as computationally simple, consisting of group‐by‐group quantile regression followed by two‐stage least squares. Using the Bahadur representation of quantile estimators, we derive weak conditions on the growth of the number of observations per group that are sufficient for consistency and asymptotic zero‐mean normality of our estimator. As in Hausman and Taylor, 1981, micro‐level covariates can be used as internal instruments for the endogenous group‐level treatment if they satisfy relevance and exogeneity conditions. Our approach applies to a broad range of settings including labor, public finance, industrial organization, urban economics, and development; we illustrate its usefulness with several such examples. Finally, an empirical application of our estimator finds that low‐wage earners in the United States from 1990 to 2007 were significantly more affected by increased Chinese import competition than high‐wage earners.     

Nonparametric Instrumental Variable Estimation of Structural Quantile Effects

We study the asymptotic distribution of Tikhonov regularized estimation of quantile structural effects implied by a nonseparable model. The nonparametric instrumental variable estimator is based on a minimum distance principle. We show that the minimum distance problem without regularization is locally ill‐posed, and we consider penalization by the norms of the parameter and its derivatives. We derive pointwise asymptotic normality and develop a consistent estimator of the asymptotic variance. We study the small sample properties via simulation results and provide an empirical illustration of estimation of nonlinear pricing curves for telecommunications services in the United States.     

Irregular Identification,Support Conditions,and Inverse Weight Estimation

In weighted moment condition models, we show a subtle link between identification and estimability that limits the practical usefulness of estimators based on these models. In particular, if it is necessary for (point) identification that the weights take arbitrarily large values, then the parameter of interest, though point identified, cannot be estimated at the regular (parametric) rate and is said to be irregularly identified. This rate depends on relative tail conditions and can be as slow in some examples as n^−1/4. This nonstandard rate of convergence can lead to numerical instability and/or large standard errors. We examine two weighted model examples: (i) the binary response model under mean restriction introduced by Lewbel (1997) and further generalized to cover endogeneity and selection, where the estimator in this class of models is weighted by the density of a special regressor, and (ii) the treatment effect model under exogenous selection (Rosenbaum and Rubin (1983)), where the resulting estimator of the average treatment effect is one that is weighted by a variant of the propensity score. Without strong relative support conditions, these models, similar to well known “identified at infinity” models, lead to estimators that converge at slower than parametric rate, since essentially, to ensure point identification, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the two models above, we derive some rates of convergence and propose that one conducts inference using rate adaptive procedures that are analogous to Andrews and Schafgans (1998) for the sample selection model.     

Local Identification of Nonparametric and Semiparametric Models

In parametric, nonlinear structural models, a classical sufficient condition for local identification, like Fisher (1966) and Rothenberg (1971), is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We derive an analogous result for the nonparametric, nonlinear structural models, establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification. Importantly, we show that additional conditions are often needed in nonlinear, nonparametric models to avoid nonlinearities overwhelming linear effects. We give restrictions on a neighborhood of the true value that are sufficient for local identification. We apply these results to obtain new, primitive identification conditions in several important models, including nonseparable quantile instrumental variable (IV) models and semiparametric consumption‐based asset pricing models.