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An invariant modification of the bilinear form test

Angelo Garate, Felipe Osorio, Federico Crudu

TL;DR

This paper addresses the noninvariance of the bilinear form BF test under reparameterization in extremum estimation. It derives sufficient conditions B1-B6 under which BF is invariant to one-to-one parameter transformations and introduces a corrected invariant statistic BF_n^c. Through Monte Carlo experiments, it shows that the corrected BF statistic achieves empirical sizes closer to the nominal level and matches the invariance properties of alternative tests like D and LM, though invariance cannot be guaranteed in all reparametrizations. The work highlights practical considerations for employing BF in nonlinear hypothesis testing and suggests avenues for further refinement using asymptotic corrections.

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.

An invariant modification of the bilinear form test

TL;DR

This paper addresses the noninvariance of the bilinear form BF test under reparameterization in extremum estimation. It derives sufficient conditions B1-B6 under which BF is invariant to one-to-one parameter transformations and introduces a corrected invariant statistic BF_n^c. Through Monte Carlo experiments, it shows that the corrected BF statistic achieves empirical sizes closer to the nominal level and matches the invariance properties of alternative tests like D and LM, though invariance cannot be guaranteed in all reparametrizations. The work highlights practical considerations for employing BF in nonlinear hypothesis testing and suggests avenues for further refinement using asymptotic corrections.

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.
Paper Structure (7 sections, 1 theorem, 15 equations, 1 table)

This paper contains 7 sections, 1 theorem, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. The bilinear form test
  3. An invariant bilinear form test
  4. Monte Carlo simulations
  5. Simulation setup
  6. Simulation results
  7. Concluding remarks

Key Result

Theorem 3.1

Let $\bm{g}:\Theta_*\to \mathbb{R}^q$ be a continuously differentiable function in $\bm{\theta}_* \in\Theta_*$ such that $\bm{g}_*(\bm{\phi}(\bm{\theta})) = \bm{0}$ if only if $\bm{g}(\bm{\theta}) =\bm{0}$. Let us consider the following assumptions: Then, the bilinear form test statistic given in Equation (eq:BF) is invariant, i.e. $BF_n(\bm{g}_*) = BF_n(\bm{g})$.

Theorems & Definitions (4)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Definition 3.2