# Inference on Partially Identified Parameters with Separable Nuisance Parameters: a Two-Stage Method

## TL;DR

This paper introduces a two-stage inference method for partially identified parameters in models with separable nuisance parameters, improving accuracy and applicability in complex econometric models.

## Contribution

It proposes a novel two-stage approach that estimates nuisance parameters separately and constructs the identified set with a refined test, valid under mild conditions.

## Key findings

- Method is asymptotically valid.
- Finite-sample properties are characterized.
- Applied to U.S. vehicle market data.

## Abstract

This paper develops a two-stage method for inference on partially identified parameters in moment inequality models with separable nuisance parameters. In the first stage, the nuisance parameters are estimated separately, and in the second stage, the identified set for the parameters of interest is constructed using a refined chi-squared test with variance correction that accounts for the first-stage estimation error. We establish the asymptotic validity of the proposed method under mild conditions and characterize its finite-sample properties. The method is broadly applicable to models where direct elimination of nuisance parameters is difficult or introduces conservativeness. Its practical performance is illustrated through an application: structural estimation of entry and exit costs in the U.S. vehicle market based on Wollmann (2018).