Inference on Directionally Differentiable Functions

Fang, Zheng; Santos, Andrés

doi:10.1093/restud/rdy049

Cited by 97 publications

(199 citation statements)

References 94 publications

(185 reference statements)

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“…From Fang and Santos (2014), on can deduce that the Hadamard directional differential of φ → Φ(φ) at…”

Section: Appendix B: Proofs Of the Results In Sectionmentioning

confidence: 99%

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Two-sample least squares projection

Pacini

2017

Econometric Reviews

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This paper investigates the problem of making inference about the coefficients in the linear projection of an outcome variable y on covariates (x, z) when data are available from two independent random samples; the first sample contains information on only the variables (y, z), while the second sample contains information on only the covariates. In this context, the validity of existing inference procedures depends crucially on the assumptions imposed on the joint distribution of (y, z, x). This paper introduces a novel characterization of the identified set of the coefficients of interest when no assumption (except for the existence of second moments) on this joint distribution is imposed. One finding is that inference is necessarily nonstandard because the function characterizing the identified set is a nondifferentiable (yet directionally differentiable) function of the data. The paper then introduces an estimator and a confidence interval based on the directional differential of the function characterizing the identified set. Monte Carlo experiments explore the numerical performance of the proposed estimator and confidence interval.

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“…From Fang and Santos (2014), on can deduce that the Hadamard directional differential of φ → Φ(φ) at…”

Section: Appendix B: Proofs Of the Results In Sectionmentioning

confidence: 99%

“…To establish theoretical properties of the bias-corrected estimator, we verify the conditions of a result in Fang and Santos (2014). For the sake of completeness, we begin by re-stating this result in a notation suitable for our purposes: …”

Section: Appendix B: Proofs Of the Results In Sectionmentioning

confidence: 99%

Two-sample least squares projection

Pacini

2017

Econometric Reviews

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“…When simulation is used to replace the population moments, this therefore could affect the behavior of the statistic in non-trivial ways. This feature motivated the vast literature on moment inequalities that corrects the critical value whenever this type of discontinuity is a concern (see e.g Andrews & Soares, 2010;Chernozhukov et al, 2013;Fang & Santos, 2014;Romano, Shaikh & Wolf, 2014).…”

Section: Regularization Of Test Statisticsmentioning

confidence: 99%

“…The non-standard nature of inference in moment inequality models and its finite-sample properties have been extensively studied in the recent literature (Andrews & Guggenberger, 2009;Andrews & Soares, 2010;Hirano & Porter, 2012;Chernozhukov et al, 2013;Fang & Santos, 2014). However, to our knowledge, its consequence in relation to simulation-based inference has not been explored.…”

Section: Simulated Moments and Motivating Examplesmentioning

confidence: 99%

Moment inequalities in the context of simulated and predicted variables

Kaido¹,

Li²,

Rysman³

2018

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This paper explores the effects of simulated moments on the performance of inference methods based on moment inequalities. Commonly used confidence sets for parameters are level sets of criterion functions whose boundary points may depend on sample moments in an irregular manner. Due to this feature, simulation errors can affect the performance of inference in non-standard ways. In particular, a (first-order) bias due to the simulation errors may remain in the estimated boundary of the confidence set. We demonstrate, through Monte Carlo experiments, that simulation errors can significantly reduce the coverage probabilities of confidence sets in small samples. The size distortion is particularly severe when the number of inequality restrictions is large. These results highlight the danger of ignoring the sampling variations due to the simulation errors in moment inequality models. Similar issues arise when using predicted variables in moment inequalities models. We propose a method for properly correcting for these variations based on regularizing the intersection of moments in parameter space, and we show that our proposed method performs well theoretically and in practice.

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“…We then do inference using an asymptotically valid bootstrap procedure. Our asymptotic results rely on important recent work by Fang and Santos (2015) and Hong and Li (2015). We construct onesided lower uniform confidence bands for the breakdown frontier.…”

Section: Introductionmentioning

confidence: 98%

Inference on breakdown frontiers

Masten¹,

Poirier²

2017

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A breakdown frontier is the boundary between the set of assumptions which lead to a specific conclusion and those which do not. In a potential outcomes model with a binary treatment, we consider two conclusions: First, that ATE is at least a specific value (e.g., nonnegative) and second that the proportion of units who benefit from treatment is at least a specific value (e.g., at least 50%). For these conclusions, we derive the breakdown frontier for two kinds of assumptions: one which indexes deviations from random assignment of treatment, and one which indexes deviations from rank invariance. These classes of assumptions nest both the point identifying assumptions of random assignment and rank invariance and the opposite end of no constraints on treatment selection or the dependence structure between potential outcomes. This frontier provides a quantitative measure of robustness of conclusions to deviations in the point identifying assumptions. We derive √ N -consistent sample analog estimators for these frontiers. We then provide an asymptotically valid bootstrap procedure for constructing lower uniform confidence bands for the breakdown frontier. As a measure of robustness, this confidence band can be presented alongside traditional point estimates and confidence intervals obtained under point identifying assumptions. We illustrate this approach in an empirical application to the effect of child soldiering on wages. We find that conclusions are fairly robust to failure of rank invariance, when random assignment holds, but conclusions are much more sensitive to both assumptions for small deviations from random assignment.JEL classification: C14; C18; C21; C25; C51

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Inference on Directionally Differentiable Functions

Cited by 97 publications

References 94 publications

Two-sample least squares projection

Two-sample least squares projection

Moment inequalities in the context of simulated and predicted variables

Inference on breakdown frontiers

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