Uniform inference for value functions

TL;DR

This paper introduces a novel approach for conducting uniform inference on value functions, overcoming non-differentiability issues by leveraging directional derivatives, and provides practical resampling techniques validated through simulations and an application to job training program evaluation.

Contribution

It develops a new inference method for value functions using directional derivatives and combines bootstrap techniques for practical implementation, addressing a key challenge in marginal optimization inference.

Findings

The proposed methods achieve accurate empirical size in simulations.

The techniques demonstrate nontrivial power in finite samples.

Application to job training data illustrates practical utility.

Abstract

We propose a method to conduct uniform inference for the (optimal) value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. Marginal optimization is not Hadamard differentiable (that is, compactly differentiable) as a map between the spaces of objective and value functions, which is problematic because standard inference methods for nonlinear maps usually rely on Hadamard differentiability. However, we show that the map from objective function to an $L_p$ functional of a value function, for $1 \leq p \leq \infty$, are Hadamard directionally differentiable. As a result, we establish consistency and weak convergence of nonparametric plug-in estimates of Cram\'er-von Mises and Kolmogorov-Smirnov test statistics applied to value functions. For practical inference, we develop detailed resampling techniques that combine a bootstrap procedure with estimates of the directional derivatives. In addition, we establish local size control of tests which use the resampling procedure. Monte Carlo simulations assess the finite-sample properties of the proposed methods and show accurate empirical size and nontrivial power of the procedures. Finally, we apply our methods to the evaluation of a job training program using bounds for the distribution function of treatment effects.