# Weak Identification Robust Tests for Subvectors Using Implied Probabilities

## TL;DR

This paper introduces a new statistical test that improves accuracy when testing parts of a model's parameters under weak identification.

## Contribution

A novel two-step test using implied probabilities is proposed to address over-rejection in weakly identified models.

## Key findings

- The new test reduces size distortion and improves finite-sample performance.
- Simulations confirm the test's strong size and power properties.
- Application to veteran earnings data shows a negative impact of veteran status.

## Abstract

This paper develops tests for hypotheses concerning subvectors of parameters in models defined by moment conditions. It is well known that conventional tests such as Wald, Likelihood-ratio and Score tests tend to over-reject when the identification is weak. To prevent uncontrolled size distortion and introduce refined finite-sample performance, we extend the projection-based test to a modified version of the score test using implied probabilities obtained by information theoretic criteria. Our test is performed in two steps, where the first step reduces the space of parameter candidates, while the second one involves the modified score test mentioned earlier. We derive the asymptotic properties of this procedure for the entire class of Generalized Empirical Likelihood implied probabilities. Simulations show that the test has very good finite-sample size and power. Finally, we apply our approach to the veteran earnings and find a negative impact of the veteran status.