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Author: Denis Chetverikov
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Languages : en
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The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the regression function is monotone and consider sieve estimators that enforce the monotonicity constraint. We define a restricted measure of ill-posedness that is relevant for the constrained estimators and show that under the monotone IV assumption and certain other conditions, our measure of ill-posedness is bounded uniformly over the dimension of the sieve space, in stark contrast with a well-known result that the unrestricted sieve measure of ill-posedness that is relevant for the unconstrained estimators grows to infinity with the dimension of the sieve space. Based on this result, we derive a novel non-asymptotic error bound for the constrained estimators. The bound gives a set of data-generating processes where the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimators. The bound shows that the regularization effect can be strong even in large samples and for steep regression functions if the NPIV model is severely ill-posed a finding that is confirmed by our simulation study. We apply the constrained estimator to the problem of estimating gasoline demand from U.S. data.
Author: Denis Chetverikov
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Languages : en
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The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the regression function is monotone and consider sieve estimators that enforce the monotonicity constraint. We define a restricted measure of ill-posedness that is relevant for the constrained estimators and show that under the monotone IV assumption and certain other conditions, our measure of ill-posedness is bounded uniformly over the dimension of the sieve space, in stark contrast with a well-known result that the unrestricted sieve measure of ill-posedness that is relevant for the unconstrained estimators grows to infinity with the dimension of the sieve space. Based on this result, we derive a novel non-asymptotic error bound for the constrained estimators. The bound gives a set of data-generating processes where the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimators. The bound shows that the regularization effect can be strong even in large samples and for steep regression functions if the NPIV model is severely ill-posed a finding that is confirmed by our simulation study. We apply the constrained estimator to the problem of estimating gasoline demand from U.S. data.
Author: Samuele Centorrino
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Languages : en
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Author: Anqi Cheng
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Languages : en
Pages : 86
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The instrumental variable approach has been widely used for estimating the treatment effect in the presence of unmeasured confounding, e.g. randomized trials with noncompliance problems and observational studies. While most literature focus on the estimation of compliers averaged causal effect (CACE) nonparametrically or based on parametric assumptions, under the IV assumptions, fewer works focus on estimating distributional causal effect using IV. We study a novel monotone cumulative distribution function estimator of an outcome variable for compliers receiving treatment or control. The estimation procedures involve a weighted quantile regression and a post-estimation rearrangement adjustment. We show that the proposed estimator is consistent and develop large sample properties. Based on the asymptotic properties of the proposed estimator, a Wilcoxon-type statistic is proposed to test the equivalence of CDF for compliers receiving treatment and control. By comparing the influence function of the proposed estimator to the efficient influence function, we modify the proposed estimator and obtain a local efficient and robust estimator in the sense that when the unknown density functions are correctly specified, it reaches the semiparametric efficiency bound and when the unknown density functions are misspecified, it is still a consistent estimator. For the censoring outcomes, we propose a method to estimate quantile functions and survival functions for potential outcomes under independent censoring and noncompliance. Based on the martingale feature associated with the censoring data, we estimate quantile functions for compliers. Then using the possibly non-monotone quantile function, we construct a monotone and bounded estimator for the survival function. By using empirical process techniques, we establish asymptotic properties, including uniform consistency and weak convergence for the proposed estimators. For general observational studies with unmeasured confounding problems, we impose a no-interaction assumption proposed by Wang and Tchetgen Tchetgen (2018) and propose a new class of IV models that identify quantities of potential outcomes for the whole population. Our work complements current research on using instrumental variable method to estimate distributions of potential outcomes and infer heterogenous treatment effect for observational studies in the presence of unmeasured confounding, especially for the censoring outcomes. Simulation results, real data examples, and proofs are detailed in this dissertation.
Author: James Joseph Heckman
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Category : Instrumental variables (Statistics)
Languages : en
Pages : 40
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This paper exposits and relates two distinct approaches to bounding the average treatment effect. One approach, based on instrumental variables, is due to Manski (1990, 1994), who derives tight bounds on the average treatment effect under a mean independence form of the instrumental variables (IV) condition. The second approach, based on latent index models, is due to Heckman and Vytlacil (1999, 2000a), who derive bounds on the average treatment effect that exploit the assumption of a nonparametric selection model with an exclusion restriction. Their conditions imply the instrumental variable condition studied by Manski, so that their conditions are stronger than the Manski conditions. In this paper, we study the relationship between the two sets of bounds implied by these alternative conditions. We show that: (1) the Heckman and Vytlacil bounds are tight given their assumption of a nonparametric selection model; (2) the Manski bounds simplify to the Heckman and Vytlacil bounds under the nonparametric selection model assumption.
Author: Victor Chernozhukov
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ISBN:
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Languages : en
Pages : 104
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Author: Markus Frölich
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Category :
Languages : en
Pages : 46
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Author: Joachim Freyberger
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Languages : en
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This paper is concerned with inference about an unidentified linear functional, L(g), where the function g satisfies the relation Y=g(x) + U; E(U/W) = 0. In this relation, Y is the dependent variable, X is a possibly endogenous explanatory variable, W is an instrument for X, and U is an unobserved random variable. The data are an independent random sample of (Y, X, W). In much applied research, X and W are discrete, and W has fewer points of support than X. Consequently, neither g nor L(g) is nonparametrically identified. Indeed, L(g) can have any value in ( -oo, oo). In applied research, this problem is typically overcome and point identification is achieved by assuming that g is a linear function of X. However, the assumption of linearity is arbitrary. It is untestable if W is binary, as is the case in many applications. This paper explores the use of shape restrictions, such as monotonicity or convexity, for achieving interval identification of L(g). Economic theory often provides such shape restrictions. This paper shows that they restrict L(g) to an interval whose upper and lower bounds can be obtained by solving linear programming problems. Inference about the identified interval and the functional L(g) can be carried out by using by using the bootstrap. An empirical application illustrates the usefulness of shape restrictions for carrying out nonparametric inference about L(g).
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Languages : en
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Author: P. Groeneboom
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ISBN: 9781139020893
Category : Estimation theory
Languages : en
Pages :
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Author: Christoph Breunig
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Languages : en
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There are many environments in econometrics which require nonseparable modeling of a structural disturbance. In a nonseparable model, key conditions are validity of instrumental variables and monotonicity of the model in a scalar unobservable. Under these conditions the nonseparable model is equivalent to an instrumental quantile regression model. A failure of the key conditions, however, makes instrumental quantile regression potentially inconsistent. This paper develops a methodology for testing the hypothesis whether the instrumental quantile regression model is correctly speci ed. Our test statistic is asymptotically normally distributed under correct speci cation and consistent against any alternative model. In addition, test statistics to justify model simpli cation are established. Finite sample properties are examined in a Monte Carlo study and an empirical illustration.
