Inference based on Kotlarski's Identity

TL;DR

This paper develops a new method for inference on latent variable densities using Kotlarski's identity, providing a confidence band with uniform size control and consistency, supported by simulations.

Contribution

It introduces a novel confidence band for the density function based on reformulating Kotlarski's identity as linear moment restrictions, addressing a two-decade open problem.

Findings

The confidence band controls asymptotic size uniformly.

The method is consistent against all fixed alternatives.

Simulation studies validate the theoretical properties.

Abstract

Kotlarski's identity has been widely used in applied economic research. However, how to conduct inference based on this popular identification approach has been an open question for two decades. This paper addresses this open problem by constructing a novel confidence band for the density function of a latent variable in repeated measurement error model. The confidence band builds on our finding that we can rewrite Kotlarski's identity as a system of linear moment restrictions. The confidence band controls the asymptotic size uniformly over a class of data generating processes, and it is consistent against all fixed alternatives. Simulation studies support our theoretical results.