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Linear regressions with endogeneity are widely used to estimate causal effects. This paper studies a statistical framework that has two common issues, endogeneity of the regressors, and heteroskedasticity that is allowed to depend on endogenous regressors, i.e., endogenous heteroskedasticity. We show that the presence of such conditional heteroskedasticity in the structural regression renders the two-stages least squares estimator inconsistent. To solve this issue, we propose sufficient conditions together with a control function approach to identify and estimate the causal parameters of interest. We establish statistical properties of the estimator, say consistency and asymptotic normality, and propose valid inference procedures. Monte Carlo simulations provide evidence of the finite sample performance of the proposed methods, and evaluate different implementation procedures. We revisit an empirical application about job training to illustrate the methods.

This paper develops a semi-parametric procedure for estimation of unconditional quantile partial effects using quantile regression coefficients. The estimator is based on an identification result showing that, for continuous covariates, unconditional quantile effects are a weighted average of conditional ones at particular quantile levels that depend on the covariates. We propose a two-step estimator for the unconditional effects where in the first step one estimates a structural quantile regression model, and in the second step a nonparametric regression is applied to the first step coefficients. We establish the asymptotic properties of the estimator, say consistency and asymptotic normality. Monte Carlo simulations show numerical evidence that the estimator has very good finite sample performance and is robust to the selection of bandwidth and kernel. To illustrate the proposed method, we study the canonical application of the Engel's curve, i.e. food expenditures as a share of income.

This paper develops bootstrap methods for practical statistical inference in panel data quantile regression models with fixed effects. We consider random-weighted bootstrap resampling and formally establish its validity for asymptotic inference. The bootstrap algorithm is simple to implement in practice by using a weighted quantile regression estimation for fixed effects panel data. We provide results under conditions that allow for temporal dependence of observations within individuals, thus encompassing a large class of possible empirical applications. Monte Carlo simulations provide numerical evidence the proposed bootstrap methods have correct finite sample properties. Finally, we provide an empirical illustration using the environmental Kuznets curve.

This paper develops a first-stage linear regression representation for the instrumental variables (IV) quantile regression (QR) model. The quantile first-stage is analogous to the least squares case, i.e., a linear projection of the endogenous variables on the instruments and other exogenous covariates, with the difference that the QR case is a weighted projection. The weights are given by the conditional density function of the innovation term in the QR structural model, conditional on the endogeneous and exogenous covariates, and the instruments as well, at a given quantile. We also show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first-stage. We then suggest inference procedures to evaluate the adequacy of instruments by evaluating their statistical significance using the first-stage result. The test is developed in an over-identification context, since consistent estimation of the weights for implementation of the first-stage requires at least one valid instrument to be available. Monte Carlo experiments provide numerical evidence that the proposed tests work as expected in terms of empirical size and power in finite samples. An empirical application illustrates that checking for the statistical significance of the instruments at different quantiles is important. The proposed procedures may be specially useful in QR since the instruments may be relevant at some quantiles but not at others.

This paper develops theoretical criteria and econometric methods to rank policy interventions in terms of welfare when individuals are loss-averse. Our new criterion for "loss aversion-sensitive dominance" defines a weak partial ordering of the distributions of policy-induced gains and losses. It applies to the class of welfare functions which model individual preferences with non-decreasing and loss-averse attitudes towards changes in outcomes. We also develop new statistical methods to test loss aversion-sensitive dominance in practice, using nonparametric plug-in estimates; these allow inference to be conducted through a special resampling procedure. Since point-identification of the distribution of policy-induced gains and losses may require strong assumptions, we extend our comparison criteria, test statistics, and resampling procedures to the partially-identified case. We illustrate our methods with a simple empirical application to the welfare comparison of alternative income support programs in the US., Comment: 62 pages, 6 page appendix, 7 figures, 2 tables

We propose a method to conduct uniform inference for the (optimal) value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. Marginal optimization is not Hadamard differentiable (that is, compactly differentiable) as a map between the spaces of objective and value functions, which is problematic because standard inference methods for nonlinear maps usually rely on Hadamard differentiability. However, we show that the map from objective function to an $L_p$ functional of a value function, for $1 \leq p \leq \infty$, are Hadamard directionally differentiable. As a result, we establish consistency and weak convergence of nonparametric plug-in estimates of Cram\'er-von Mises and Kolmogorov-Smirnov test statistics applied to value functions. For practical inference, we develop detailed resampling techniques that combine a bootstrap procedure with estimates of the directional derivatives. In addition, we establish local size control of tests which use the resampling procedure. Monte Carlo simulations assess the finite-sample properties of the proposed methods and show accurate empirical size and nontrivial power of the procedures. Finally, we apply our methods to the evaluation of a job training program using bounds for the distribution function of treatment effects., Comment: 64 pages, 4 figures, 8 tables

Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out individual effects, and inference is usually justified in a `large n large T' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we show that this gap can be bridged and establish asymptotic unbiased normality for quantile regression panels under conditions on n,T that are very close to what is typically assumed in standard nonlinear panels. Our results considerably improve upon existing theory and show that quantile regression is applicable to the same type of panel data (in terms of n,T) as other commonly used nonlinear panel data models. Thorough numerical experiments confirm our theoretical findings.

This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on smoothed sample moments. We propose a method of moments estimator for exactly identified models, as well as a generalized method of moments estimator for over-identified models. We establish consistency and asymptotic normality of both estimators under general conditions that allow for weakly dependent data and nonlinear structural models. Simulations illustrate the finite-sample properties of the methods. Our in-depth empirical application concerns the consumption Euler equation derived from quantile utility maximization. Advantages of the quantile Euler equation include robustness to fat tails, decoupling of risk attitude from the elasticity of intertemporal substitution, and log-linearization without any approximation error. For the four countries we examine, the quantile estimates of discount factor and elasticity of intertemporal substitution are economically reasonable for a range of quantiles above the median, even when two-stage least squares estimates are not reasonable.

This paper considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, $n$, and the number of time periods, $T$, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarded as a pseudo-true parameter. We then establish the asymptotic distributional properties of the FE estimator around the pseudo-true parameter when $n$ and $T$ jointly go to infinity. Notably, we show that the FE estimator suffers from the incidental parameters bias of which the top order is $O(T^{-1})$, and even after the incidental parameters bias is completely removed, the rate of convergence of the FE estimator depends on the degree of model misspecification and is either $(nT)^{-1/2}$ or $n^{-1/2}$. Second, we establish asymptotically valid inference on the (pseudo-true) parameter. Specifically, we derive the asymptotic properties of the clustered covariance matrix (CCM) estimator and the cross section bootstrap, and show that they are robust to model misspecification. This establishes a rigorous theoretical ground for the use of the CCM estimator and the cross section bootstrap when model misspecification and the incidental parameters bias (in the coefficient estimate) are present. We conduct Monte Carlo simulations to evaluate the finite sample performance of the estimators and inference methods, together with a simple application to the unemployment dynamics in the U.S., Comment: 53 pages; minor corrections made. To appear in JBES