An invariant modification of the bilinear form test

TL;DR

This paper proposes a simple modification to the bilinear form test to ensure invariance under reparametrization, improving its robustness in extremum estimation scenarios, supported by simulation evidence.

Contribution

It introduces conditions for invariance of the bilinear form test and a modified test statistic that maintains invariance under reparametrization.

Findings

Modified test shows good performance in simulations.

Ensures invariance under reparametrization.

Applicable to linear and nonlinear hypotheses.

Abstract

The invariance properties of certain likelihood-based asymptotic tests as well as their extensions for M-estimation, estimating functions and the generalized method of moments have been well studied. The simulation study reported in Crudu and Osorio [Econ. Lett. 187: 108885, 2020] shows that the bilinear form test is not invariant to one-to-one transformations of the parameter space. This paper provides a set of suitable conditions to establish the invariance property under reparametrization of the bilinear form test for linear or nonlinear hypotheses that arise in extremum estimation which leads to a simple modification of the test statistic. Evidence from a Monte Carlo simulation experiment suggests good performance of the proposed methodology.