Intersection bounds: estimation and inference

By Victor Chernozhukov, S. Lee, A. Rosen. 2009

456 citations recorded by the source corpus.

We develop a practical and novel method for inference on intersection bounds, namely bounds defined by either the infimum or supremum of a parametric or nonparametric function, or equivalently, the value of a linear programming problem with a potentially infinite constraint set. Our approach is especially convenient in models comprised of a continuum of inequalities that are separable in parameters, and also applies to models with inequalities that are non-separable in parameters. Since analog estimators for intersection bounds can be severely biased in finite samples, routinely underestimating the length of the identified set, we also offer a (downward/upward) median unbiased estimator of these (upper/lower) bounds as a natural by-product of our inferential procedure. Furthermore, our method appears to be the first and currently only method for inference in nonparametric models with a continuum of inequalities. We develop asymptotic theory for our method based on the strong approximation of a sequence of studentized empirical processes by a sequence of Gaussian or other pivotal processes. We provide conditions for the use of nonparametric kernel and series estimators, including a novel result that establishes strong approximation for general series estimators, which may be of independent interest. We illustrate the usefulness of our method with Monte Carlo experiments and an empirical example.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/400a7a92b52c56cabb57fe0b27a5db59ef467e16

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Citations in the corpus

Inference on Directionally Differentiable Functions (2014, unknown): “(cid:9)”
Partial Identification of Local Average Treatment Effects With an Invalid Instrument (2013, methodology): “The additional testable implication follows from combining these inequalities with equation (9) and the corresponding equation for the observed group (Z,D) = (1, 1), which is a mixture of the at and c strata.”
Partial Distributional Policy Effects (2011, unknown): “The latter can be estimated by either the parametric methods discussed in Chernozhukov et al. (2009a), e.g. by first estimating a linear quantile regression model QY |X(τ, x) = x ′β(τ), and then inverting the corresponding conditional quantile function, or by a fully nonparametric CDF…”
Gaussian approximation of suprema of empirical processes (2012, methodology): “Another way is to use a bound on the (1 − α)-quantile of W̃n using sharp deviation inequalities available to Gaussian processes, which leads to analytic construction of confidence bands; see, for example, [16] for this approach.”
Partial identification using random set theory (2010, methodology): “For the case of conditional Aumann expectations as in Propositions C.3 and C.4, estimation and statistical inference can be carried out using the methods proposed by Andrews and Shi (2009), Chernozhukov, Lee, and Rosen (2009) and Ponomareva (2010).”
Some new asymptotic theory for least squares series: Pointwise and uniform results (2012, methodology): “The first result on inference is concerned with the strong approximation of a series process by a Gaussian process and is a (relatively) minor extension of the result obtained by Chernozhukov et al. (2009).”
TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES (2013, methodology): “AS compared their tests with (Chernozhukov et al., 2013, hereafter CLR) and Lee et al. (2013).”
Fuzzy Differences-in-Differences (2015, background): “ws 4 A similar conclusion holds if y or y are estimated rather than known by the researcher: the estimators are n rather than root-n consistent and not asymptotically normal. 17 & Soares, 2010 or Chernozhukov et al., 2013). Also, it is unclear whether the condence interval CI 1 resulting from th”
Inference in Regression Discontinuity Designs with a Discrete Running Variable (2016, methodology): “ables, which can easily be simulated. Inverting these tests leads to the left-sided CI of the form [K^ 1 ;1 ), where K^ 1 solves max k;q 2f ;+ g j ^ q ks =V (^ q ks ) 1 = 2 j = q 1 (K^ 1 ). Following Chernozhukov et al. (2013), we use K^ 1 = 2 as a point estimate, which is half-median unbiased in th”
Sensitivity Analysis in Semiparametric Likelihood Models (2011, background): “…(1981) establishes consistency of the parametric MLE without assuming that it is uniquely identified, and Liu and Shao (2003) obtain the 2See also the papers of Rosen (2008), Bugni (2010), Canay (2010), Chernozhukov, Lee, and Rosen (2009), Andrews and Shi (2010) and other papers referenced therein.”
Revealed Preferences in a Heterogeneous Population (2014, background): “This ordering does not have much economic significance.(9) In particular, it is hard to see how an ordering of subjects on Bs that depends on time t prices (and vice versa) would arise from natural restrictions on behavior.”
Impossibility Results for Nondifferentiable Functionals (2012, background): “Moreover, Doss and Sethuraman (1989) show that in cases where no unbiased estimator exists, a sequence of estimators whose bias approaches zero must have variance approaching infinity. Hence, attempting to make the bias very small may lead to highly variable estimates. This does not preclude modifyi”
Testing Many Moment Inequalities (2013, methodology): “The selection procedure used in the theorem above is most closely related to those in Chernozhukov, Lee, and Rosen (2013) and in Chetverikov (2011).”
Adaptive test of conditional moment inequalities (2011, methodology): “The test of Chernozhukov et al. (2009) also has high power in a large class of CMI models but it requires knowledge of certain smoothness properties of moment functions such as order of differentiability whereas the test developed in this paper does not.”
Doubly robust uniform confidence band for the conditional average treatment effect function (2016, background): “For example, see Claeskens and Keilegom (2003) for smoothed bootstrap confidence bands and Chernozhukov, Lee, and Rosen (2013) for multiplier bootstrap confidence bands.”
Conditional quantile processes based on series or many regressors (2011, methodology): “Our strategy is related to that used in [25] for the problem of density estimation and is built on [30], who proposed a related strategy for inference on the minimum of a function.”
Set Identified Linear Models (2011, methodology): “The gain of the direct approach that we used with respect to the approach followed by Chernozhukov et al. (2007) using a criterion is an interesting question.”