Nonparametric instrumental variable estimation under monotonicity

TL;DR

This paper demonstrates that imposing monotonicity constraints in nonparametric instrumental variable models significantly improves estimation rates by reducing ill-posedness, leading to faster convergence and better finite-sample performance.

Contribution

The paper introduces a novel approach that leverages monotonicity to weaken ill-posedness in nonparametric IV estimation, enabling polynomial convergence rates regardless of ill-posedness degree.

Findings

Monotonicity constraints lead to bounded ill-posedness measures.

Constrained estimators achieve polynomial convergence rates.

Simulation shows improved finite-sample performance with monotonicity.

Abstract

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.