This paper studies the identifying power of an instrumental variable in the nonparametric heterogeneous treatment effect framework when a binary treatment is mismeasured and endogenous. Using a binary instrumental variable, I characterize the sharp identified set for the local average treatment effect under the exclusion restriction of an instrument and the deterministic monotonicity of the true treatment in the instrument. Even allowing for general measurement error (e.g., the measurement error is endogenous), it is still possible to obtain finite bounds on the local average treatment effect. Notably, the Wald estimand is an upper bound on the local average treatment effect, but it is not the sharp bound in general. I also provide a confidence interval for the local average treatment effect with uniformly asymptotically valid size control. Furthermore, I demonstrate that the identification strategy of this paper offers a new use of repeated measurements for tightening the identified set.

Single-agent dynamic discrete choice models are typically estimated using heavily parametrized econometric frameworks, making them susceptible to model misspecification. This paper investigates how misspecification affects the results of inference in these models. Specifically, we consider a local misspecification framework in which specification errors are assumed to vanish at an arbitrary and unknown rate with the sample size. Relative to global misspecification, the local misspecification analysis has two important advantages. First, it yields tractable and general results. Second, it allows us to focus on parameters with structural interpretation, instead of "pseudo-true" parameters.We consider a general class of two-step estimators based on the K-stage sequential policy function iteration algorithm, where K denotes the number of iterations employed in the estimation. This class includes Hotz and Miller (1993)'s conditional choice probability estimator, Aguirregabiria and Mira (2002)'s pseudo-likelihood estimator, and Pesendorfer and Schmidt-Dengler (2008)'s asymptotic least squares estimator.We show that local misspecification can affect the asymptotic distribution and even the rate of convergence of these estimators. In principle, one might expect that the effect of the local misspecification could change with the number of iterations K. One of our main findings is that this is not the case, that is, the effect of local misspecification is invariant to K. In practice, this means that researchers cannot eliminate or even alleviate problems of model misspecification by choosing K.We thank the three anonymous referees for comments and suggestions that have significantly improved this paper. We are also grateful for helpful discussions with

This paper studies non-separable models with a continuous treatment when the dimension of the control variables is high and potentially larger than the effective sample size. We propose a three-step estimation procedure to estimate the average, quantile, and marginal treatment effects. In the first stage we estimate the conditional mean, distribution, and density objects by penalized local least squares, penalized local maximum likelihood estimation, and numerical differentiation, respectively, where control variables are selected via a localized method of L 1 -penalization at each value of the continuous treatment. In the second stage we estimate the average and marginal distribution of the potential outcome via the plug-in principle. In the third stage, we estimate the quantile and marginal treatment effects by inverting the estimated distribution function and using the local linear regression, respectively. We study the asymptotic properties of these estimators and propose a weighted-bootstrap method for inference. Using simulated and real datasets, we demonstrate that the proposed estimators perform well in finite samples.Econometricians observe an outcome Y , a continuous treatment T , and a set of covariates X, which may be high-dimensional. They are connected by a measurable function Γ(·), i.e.,where A is an unobservable random vector and may not be weakly separable from observables (T, X), and Γ may not be monotone in either T or A.Let Y (t) = Γ(t, X, A). We are interested in the average EY (t), the marginal distribution P(Y (t) ≤ u) for some u ∈ , and the quantile q τ (t), where we denote q τ (t) as the τ -th quantile of Y (t) for some τ ∈ (0, 1). We are also interested in the causal effect of moving T from t to t , i.e., E(Y (t) − Y (t )) and q τ (t) − q τ (t ). Last, we are interested in the average marginal effect E[∂ t Γ(t, X, A)] and quantile partial derivative ∂ t q τ (t). Next, we specify conditions under which the above parameters are identified. Assumption 1The random variables A and T are conditionally independent given X.Assumption 1 is known as the unconfoundedness condition, which is commonly assumed in the treatment effect literature. See Cattaneo (2010), Cattaneo and Farrell (2011), Hirano et al. (2003) and Firpo (2007) for the case of discrete treatment and Graham, Imbens, and Ridder (2014), Galvao and Wang (2015), and Hirano and Imbens (2004) for the case of continuous treatment. It is also called the conditional independence assumption in Hoderlein and Mammen (2007), which is weaker than the full joint independence between A and (T, X). Note that X can be arbitrarily correlated with the unobservables A. This assumption is more plausible when we control for sufficiently many and potentially high-dimensional covariates.Theorem 2.1 Suppose Assumption 1 holds and Γ(·) is differentiable in its first argument. Then the marginal distribution of Y (t) and the average marginal effect ∂ t EY (t) are identified. In addition, if Assumption 6 in the Appendix holds and X is continuously distribute...

Researchers often trim observations with small values of the denominator A when they estimate moments of the form $\mathbb {E}[B/A]$ . Large trimming is common in practice to reduce variance, but it incurs a large bias. This paper provides a novel method of correcting the large trimming bias. If a researcher is willing to assume that the joint distribution between A and B is smooth, then the trimming bias may be estimated well. Along with the proposed bias correction method, we also develop an inference method. Practical advantages of the proposed method are demonstrated through simulation studies, where the data generating process entails a heavy-tailed distribution of $B/A$ . Applying the proposed method to the Compustat database, we analyze the history of external financial dependence of U.S. manufacturing firms for years 2000–2010.

The literature on dynamic discrete games often assumes that the conditional choice probabilities and the state transition probabilities are homogeneous across markets and over time. We refer to this as the "homogeneity assumption" in dynamic discrete games. This homogeneity assumption enables empirical studies to estimate the game's structural parameters by pooling data from multiple markets and from many time periods. In this paper, we propose a hypothesis test to evaluate whether the homogeneity assumption holds in the data. Our hypothesis is the result of an approximate randomization test, implemented via a Markov chain Monte Carlo (MCMC) algorithm. We show that our hypothesis test becomes valid as the (user-defined) number of MCMC draws diverges, for any fixed number of markets, time-periods, and players.We apply our test to the empirical study of the U.S. Portland cement industry in Ryan (2012).