THE QUANTITIES DISCUSSED in Sections 2.2 and 2.3 are well-defined and have causal interpretation under standard conditions. We briefly recall these conditions, using the potential outcomes notation. Let Y u1 and Y u0 denote the potential outcomes under the treatment states 1 and 0. These outcomes are not observed jointly, and we instead observe Exogeneity fails when D depends on the potential outcomes. For example, people may drop out of a program if they think the program will not benefit them. In this case, instrumental variables are useful in creating quasi-experimental fluctuations in D that may identify useful effects. Let Z be a binary instrument, such as an offer of participation, that generates potential participation decisions D 1 and D 0 under the instrument states 1 and 0, respectively. As with the potential outcomes, the potential participation decisions under both instrument states are not observed jointly. The realized participation decision is then given by D = ZD 1 + (1 − Z)D 0 . We assume that Z is assigned randomly with respect to potential outcomes and participation decisions conditional on X, that is,There are many causal quantities of interest for program evaluation. Chief among these are various structural averages:, the causal LASF-T; as well as effects derived from them such as, the causal LATE-T. These causal quantities are the same as the structural parameters defined in Sections 2.2-2.3 under the following well-known sufficient condition. ASSUMPTION G.1-Assumptions for Causal/Structural Interpretability: The following conditions hold P-almost surely:This condition due to Imbens and Angrist (1994) and Abadie (2003) is much-used in the program evaluation literature. It has an equivalent formulation in terms of a simultaneous

We propose strategies to estimate and make inference on key features of heterogeneous effects in randomized experiments. These key features include best linear predictors of the effects using machine learning proxies, average effects sorted by impact groups, and average characteristics of most and least impacted units. The approach is valid in high dimensional settings, where the effects are proxied by machine learning methods. We post-process these proxies into the estimates of the key features. Our approach is generic, it can be used in conjunction with penalized methods, deep and shallow neural networks, canonical and new random forests, boosted trees, and ensemble methods. It does not rely on strong assumptions. In particular, we don't require conditions for consistency of the machine learning methods. Estimation and inference relies on repeated data splitting to avoid overfitting and achieve validity. For inference, we take medians of p-values and medians of confidence intervals, resulting from many different data splits, and then adjust their nominal level to guarantee uniform validity. This variational inference method is shown to be uniformly valid and quantifies the uncertainty coming from both parameter estimation and data splitting. We illustrate the use of the approach with two randomized experiments in development on the effects of microcredit and nudges to stimulate immunization demand.

We derive fixed effects estimators of parameters and average partial effects in (possibly dynamic) nonlin-ear panel data models with individual and time effects. They cover logit, probit, ordered probit, Poisson and Tobit models that are important for many empirical applications in micro and macroeconomics. Our estimators use analytical and jackknife bias corrections to deal with the incidental parameter problem, and are asymptotically unbiased under asymptotic sequences where N/T converges to a constant. We develop inference methods and show that they perform well in numerical examples.

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Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data.We develop large sample theory for the QR-series coefficient process, namely we obtain uniform strong approximations to the QR-series coefficient process by conditionally pivotal and Gaussian processes. Based on these two strong approximations, or couplings, we develop four resampling methods (pivotal, gradient bootstrap, Gaussian, and weighted bootstrap) that can be used for inference on the entire QR-series coefficient function.We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives.Specifically, we obtain uniform rates of convergence and show how to use the four resampling methods mentioned above for inference on the functionals. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and the covariate value, and covering the pointwise case as a by-product. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function and test the Slutsky condition of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption. and Yale for many useful suggestions. We are grateful to Adonis Yatchew for giving us permission to use the data set in the empirical application. We gratefully acknowledge research support from the NSF. The R package quantreg.nonpar implements some of the methods of this paper [66]. 1 arXiv:1105.6154v4 [stat.ME] 9 Aug 2018 for a careful and detailed description of these series functions; see also Belloni et al [10] for an overview of recent advances on series approximating functions.We define the QR-series approximating function x → Z(x) β(u) mapping X into R, and, for all x ∈ X , the QR-series approximation error2 The optimization problem (2.2) has a finite solution if E[|Q(u, X)|] is finite. In addition, since the function z → ρu(z) is strictly convex, the solution is unique if the matrix E[Z(X)Z(X) ] is non-singular, which is assumed in Condition S below. The term ρu(Y − Q(u, X)) does not affect the optimization problem but guarantees the existence of the solution when E[|Y |] is not finite.3 Interestingly, in the case of B-splines and compactly supported wavelets, the entire collection of series terms is dependent upon th...

We thank Denis Chetverikov and Sukjin Han for excellent comments and capable research assistance. We are grateful to Richard Blundell for providing us the data for the empirical application. Stata software to implement the methods developed in the paper is available in Amanda Kowalski's web site at http://www.econ.yale.edu/ak669/research.html. We gratefully acknowledge research support from the NSF. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.