Nonparametric instrumental regression with non-convex constraints

TL;DR

This paper develops a Tikhonov regularization-based estimator for nonparametric instrumental regression models with non-convex shape constraints, providing convergence rates under specific source and smallness conditions.

Contribution

It introduces a novel estimator for nonparametric instrumental regression with non-convex constraints and derives convergence rates in both deterministic and stochastic settings.

Findings

Convergence rates are established for the proposed estimator.

The method handles non-convex shape restrictions effectively.

Theoretical analysis includes deterministic and stochastic scenarios.

Abstract

This paper considers the nonparametric regression model with an additive error that is dependent on the explanatory variables. As is common in empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. A classical example in microeconomics considers the consumer demand function as a function of the price of goods and the income, both variables often considered as endogenous. In this framework, the economic theory also imposes shape restrictions on the demand function, like integrability conditions. Motivated by this illustration in microeconomics, we study an estimator of a nonparametric constrained regression function using instrumental variables by means of Tikhonov regularization. We derive rates of convergence for the regularized model both in a deterministic and stochastic setting under the assumption that the true regression function satisfies a projected source condition including, because of the non-convexity of the imposed constraints, an additional smallness condition.