Testing regression monotonicity in econometric models

Chetverikov, Denis

doi:10.1920/wp.cem.2012.3512

Cited by 31 publications

(32 citation statements)

References 70 publications

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“…It is also possible to extend our results to show that Theorems 5 and 6 hold with = 0 at the expense of additional technicalities. Further, using the same arguments as those in Chetverikov (2012), it is possible to show that the test suggested here has minimax optimal rate of consistency against the alternatives belonging to certain Hölder classes for a reasonably large range of smoothness levels. We do not derive these results here for the sake of brevity of presentation.…”

Section: Remark 18 (Weak Condition On the Bandwidth Values) Our Theomentioning

confidence: 83%

“…. , p. In turn, Proposition B.2 in Chetverikov (2012) shows that under Assumptions 2, 9, and 10, with probability at least 1 − Cn −c , z ij ≤ C/ √ h min =: B n uniformly over all j = 1, . .…”

Section: Proofs For Sectionmentioning

confidence: 96%

“…An adaptive test of this hypothesis was developed in Chetverikov (2012). In our case, H 0 requires the regression function w → E[1{X ≤ x}|W = w] to be decreasing not only for a particular value x ∈ (0, 1) but for all x ∈ (0, 1), and so we need to extend the results obtained in Chetverikov (2012).…”

Section: Testing the Monotonicity Assumptionsmentioning

confidence: 99%

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Nonparametric instrumental variable estimation under monotonicity

Wilhelm,

Chetverikov

2016

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The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression functions and monotone instruments significantly weakens the ill-posedness of the problem. In stark contrast to the existing literature, the presence of a monotone instrument implies boundedness of our measure of ill-posedness when restricted to the space of monotone functions. Based on this result we derive a novel non-asymptotic error bound for the constrained estimator that imposes monotonicity of the regression function. For a given sample size, the bound is independent of the degree of ill-posedness as long as the regression function is not too steep. As an implication, the bound allows us to show that the constrained estimator converges at a fast, polynomial rate, independently of the degree of ill-posedness, in a large, but slowly shrinking neighborhood of constant functions. Our simulation study demonstrates significant finite-sample performance gains from imposing monotonicity even when the regression function is rather far from being a constant. We apply the constrained estimator to the problem of estimating gasoline demand functions from U.S. data. *

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Section: Remark 18 (Weak Condition On the Bandwidth Values) Our Theomentioning

confidence: 83%

Section: Proofs For Sectionmentioning

confidence: 96%

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Nonparametric instrumental variable estimation under monotonicity

Wilhelm,

Chetverikov

2016

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“…Our general framework is also related to testing qualitative nonparametric hypotheses such as monotonicity in mean regression. See, for example, Baraud, Huet, and Laurent (2005), Chetverikov (2012), Dümbgen and Spokoiny (2001), and Ghosal, Sen, and van der Vaart (2000) among many others. See also Lee, Linton, and Whang (2009) and Delgado and Escanciano (2012) for testing stochastic monotonicity.…”

Section: Introductionmentioning

confidence: 99%

Testing for a general class of functional inequalities

Lee¹,

Song²,

Whang³

2014

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Abstract. In this paper, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of L p functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating "contact sets." We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including the cases that the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.

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“…Our result on the anti-concentration of separable Gaussian processes is also of independent interest in many other problems. For example, applications of our anti-concentration bounds can be found in [9] and [10], which consider the problems of nonparametric inference on a minimum of a function and nonparametric testing of qualitative hypotheses about functions, respectively. 1.1.…”

Section: Introductionmentioning

confidence: 99%

Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

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Abstract. Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, . This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value.We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Furthermore, our approach is asymptotically honest at a polynomial rate -namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate -the error converges to zero at a slow, logarithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

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Testing regression monotonicity in econometric models

Cited by 31 publications

References 70 publications

Nonparametric instrumental variable estimation under monotonicity

Nonparametric instrumental variable estimation under monotonicity

Testing for a general class of functional inequalities

Anti-concentration and honest, adaptive confidence bands

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