On the Asymptotic Distribution of Variance Weighted KS Statistics

TL;DR

This paper derives the asymptotic distribution of variance weighted KS statistics for conditional moment inequality models with a one-dimensional covariate, enabling analytical critical value calculation and optimal local power.

Contribution

It provides the first derivation of the asymptotic distribution for these statistics, improving test efficiency and simplicity in one-dimensional settings.

Findings

Asymptotic distribution depends only on the variance of a single variable.

Critical values can be calculated analytically.

Tests achieve minimax rate for local alternatives.

Abstract

This paper derives the asymptotic distribution of variance weighted Kolmogorov-Smirnov statistics for conditional moment inequality models for the case of a one dimensional covariate. The asymptotic distribution depends on the data generating process only through the variance of a single random variable, leading to critical values that can be calculated analytically. By arguments in Armstrong (2011b), the resulting tests achieve the best minimax rate for local alternatives out of available approaches in a broad class of settings.