Denis Chetverikov, Daniel Wilhelm
arXiv 19 Jul 2015 · Statistics — Applications · publishedEconometrica (2017) · 41 citations (OpenAlex)
arXiv:1507.05270 · PDF · DOI · OpenAlex · Extracted main text
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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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Lee, Linton, and Whang (2009) Testing for Stochastic Monotonicity | 1.000 | 10 | 3 | 100% |
| 2 | Horowitz and Lee (2012) Uniform confidence bands for functions estimated nonparametrically with instrumental variables | 0.928 | 4 | 3 | 100% |
| 3 | Blundell, Chen, and Kristensen (2007) Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves | 0.874 | 12 | 2 | 100% |
| 4 | Horowitz (2012) Specification Testing in Nonparametric Instrumental Variable Estimation | 0.874 | 6 | 2 | 100% |
| 5 | Chetverikov (2012) Testing Regression Monotonicity in Econometric Models self | 0.763 | 9 | 2 | 67% |
| 6 | Belloni, Chernozhukov, Chetverikov, and Kato (2014) Some New Asymptotic Theory for Least Squares Series: Pointwise and Uniform Results | 0.737 | 3 | 3 | 67% |
| 7 | Mammen (1991) Estimating a Smooth Monotone Regression Function | 0.737 | 3 | 2 | 100% |
| 8 | Blundell, Horowitz, and Parey (2012) Measuring the price responsiveness of gasoline demand: Economic shape restrictions and nonparametric demand estimation | 0.693 | 5 | 1 | 100% |
| 9 | Blundell, Horowitz, and Parey (2013) Nonparametric Estimation of a Heterogeneous Demand Function under the Slutsky Inequality Restriction | 0.644 | 2 | 2 | 100% |
| 10 | Horowitz (2014) Ill-Posed Inverse Problems in Economics | 0.644 | 2 | 2 | 100% |
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