An invariant modification of the bilinear form test

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.

💡 Research Summary

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The paper addresses a crucial shortcoming of the bilinear form (BF) test introduced by Crudu and Osorio (2020) for extremum estimation: the test is not invariant to one‑to‑one reparameterizations of the parameter space. Invariance is a desirable property because statistical conclusions should not depend on the particular algebraic representation of the null hypothesis. The authors first revisit the BF test within the general extremum‑estimation framework, where an objective function (Q_n(\theta)) is maximized (or minimized) over (\theta\in\Theta\subset\mathbb{R}^p). The null hypothesis is expressed as a set of (q) nonlinear restrictions (g(\theta)=0). Under regularity conditions (A1–A2) the BF statistic

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