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An invariant modification of the bilinear form test
\address{Departamento de Estad\'istica, Pontificia Universidad Cat\'olica de Chile} \curraddr{Avenida Vicu\ na Mackena 4860, Santiago, Chile} \email{[email removed]}
\email{[email removed]}
\address{Department of Economics and Statistics, University of Siena, Italy} \curraddr{Piazza San Francesco, 7/8 53100 Siena, Italy.} \email{[email removed]}
Recently, Dufour:2017 have studied the invariance properties of various testing procedures for nonlinear hypotheses in the context of $M$-estimation, inference functions and the generalized method of moments (GMM) to equivalent reformulation and reparametrization of the hypotheses of interest. The main conclusion of their study is that as in the likelihood framework Dagenais:1991, the Lagrange multiplier-type statistics, $C(\alpha)$-type and the distance metric criterion proposed by Newey:1987 are invariant to reformulations of the null hypothesis and to one-to-one transformations of the parameter space, whereas the Wald-type statistic does not possess such properties. Although the Wald-type statistic cannot be recommended in such circumstances, some solutions Kemp:2001 have been proposed to make the test invariant to general reparametrizations.
It is worth emphasizing that the estimation mechanisms considered by Dufour:2017 belong to the class of extremum estimators Gourieroux:1995, Hayashi:2000. Besides, they only consider test statistics that correspond to extensions of the so-called “holy trinity" Muggeo:2014. By contrast, Crudu:2020 proposed a competitive procedure for testing nonlinear hypotheses, based on a bilinear form (BF) statistic, in the context of extremum estimation that generalizes the gradient statistic of Terrell:2002. As stated in Rao:2005, additional research is needed to better understand the behavior of the gradient statistic. This has prompted a series of works regarding various aspects of the asymptotic behavior of the gradient statistic Lemonte:2013. The simulation study in Crudu:2020 revealed that the BF test lacks invariance under equivalent null hypotheses. The main aim of this work is to provide suitable conditions to ensure the invariance under reparametrization property of the BF test based on extremum estimation.
This paper is organized as follows. In Section (ref), the bilinear form test for extremum estimation is revisited. Section (ref) provides sufficient conditions for the invariance of the bilinear form test statistic. In Section (ref), we report our numerical findings that allow us to illustrate the invariance to one-to-one transformations of the parameter space, i.e., reparametrizations. Finally some concluding remarks are discussed in Section (ref).
The general class of estimating procedures known as extremum estimation Gourieroux:1995 is defined by optimizing an objective function $Q_n(\mbox{\boldmath $\theta$})$, which depends on observed data $\mbox{\boldmath $z$}_1,\dots,\mbox{\boldmath $z$}_n$ and a parameter vector $\mbox{\boldmath $\theta$}\in\Theta\subset\mathbb{R}^p$, which includes, for example, GMM Hansen:1982, $M$-estimation Huber:1981, and maximum likelihood estimation under model misspecification White:1982. In Crudu:2020, the BF statistic was introduced to test nonlinear hypotheses of the form,
where $\mbox{\boldmath $g$}:\Theta\to \mathbb{R}^q$ is a continuously differentiable function of $\mbox{\boldmath $\theta$}$ and $\mbox{\boldmath $G$}(\mbox{\boldmath $\theta$}) = \partial\mbox{\boldmath $g$}(\mbox{\boldmath $\theta$})/\partial\mbox{\boldmath $\theta$}^\top$ is a $q\times p$ matrix with full row rank. Define $\mbox{\boldmath $A$}_n(\mbox{\boldmath $\theta$}) = \partial^2 Q_n(\mbox{\boldmath $\theta$})/\partial\mbox{\boldmath $\theta$}\partial\mbox{\boldmath $\theta$}^\top$. The following regularity conditions are assumed:
Then, the BF statistic for testing the hypothesis defined in (ref) is given by
where $\mbox{\boldmath $G$} = \mbox{\boldmath $G$}(\mbox{\boldmath $\theta$})$, $\mbox{\boldmath $S$} = \mbox{\boldmath $G$}(-\mbox{\boldmath $A$})^{-1}\mbox{\boldmath $G$}^\top$, $\mbox{\boldmath $\Omega$} = \mbox{\boldmath $GA$}^{-1}\mbox{\boldmath $BA$}^{-1}\mbox{\boldmath $G$}^\top$ with $\mbox{\boldmath $G$}^+ = \mbox{\boldmath $G$}^\top (\mbox{\boldmath $GG$}^\top)^{-1}$ being the Moore-Penrose inverse of $\mbox{\boldmath $G$}$, $\widehat{\mbox{\boldmath $\theta$}}_n$ is the unrestricted extremum estimator and the constrained estimator $\widetilde{\mbox{\boldmath $\theta$}}_n$ is the solution to the problem: \[ \max_{\theta\in\Theta}\ Q_n(\mbox{\boldmath $\theta$}), \qquad \text{subject to: $\mbox{\boldmath $g$}(\mbox{\boldmath $\theta$}) =\mbox{\boldmath $0$}$}. \] Given Assumptions A1-A2 and under $H_0$, the $BF_n(\mbox{\boldmath $g$})$ statistic given in ((ref)) asymptotically has a chi-square distribution with $q$ degrees of freedom. Details about the asymptotic distribution for the BF statistic and its asymptotic equivalence with the Lagrange multiplier test statistic can be found in Crudu:2020.
Let us consider a reparameterization of the parameter space Dufour:2017 defined by a one-to-one differentiable transformation $\mbox{\boldmath $\phi$}:\Theta\to\Theta_*$ with $\Theta\subset\mathbb{R}^p$ and $\Theta_*\subset\mathbb{R}^p$ such that $\mbox{\boldmath $\phi$}(\mbox{\boldmath $\theta$}) = \mbox{\boldmath $\theta$}_*$ and the inverse function of $\mbox{\boldmath $\phi$}$ satisfies $\mbox{\boldmath $\phi$}^{-1}(\mbox{\boldmath $\theta$}_*) = \mbox{\boldmath $\theta$}$. Suppose that the following holds
then, the null hypotheses $H_0:\mbox{\boldmath $g$}(\mbox{\boldmath $\theta$}) = \mbox{\boldmath $0$}$ and $H_0^*:\mbox{\boldmath $g$}_*(\mbox{\boldmath $\theta$}_*) = \mbox{\boldmath $0$}$ are considered equivalent representations of the same hypothesis, provided that $\mbox{\boldmath $g$}(\mbox{\boldmath $\theta$}) = \mbox{\boldmath $0$}$ if and only if $\mbox{\boldmath $g$}_*(\mbox{\boldmath $\theta$}_*) = \mbox{\boldmath $0$}$, in which case we say that the test is invariant to reparametrization Dufour:2017. The following theorem states sufficient conditions for the invariance to reparametrization for the BF test.
Clearly, conditions B4 and B5 can be disregarded when $\mbox{\boldmath $B$} = -\mbox{\boldmath $A$}$ which occurs, for example, when the objective function $Q_n(\mbox{\boldmath $\theta$})$ is based on a quasi-score estimating equation. This is particularly relevant in the context of GMM, where it is beneficial to use an invariant test that avoids computing matrices $\mbox{\boldmath $A$}$ and $\mbox{\boldmath $B$}$.
It is worth noting that Greville:1966 highlights the conditions necessary to validate assumptions like the one stated in B2. To establish B2, it is sufficient to verify the following identity Greville:1966,
Here, $\mbox{\boldmath $G$}$ is a full row rank matrix, which implies that its Moore-Penrose inverse is given by $\mbox{\boldmath $G$}^+ = \mbox{\boldmath $G$}^\top(\mbox{\boldmath $GG$}^\top)^{-1}$. This expression facilitates the verification of identity (ref). Additionally, $\mbox{\boldmath $K$}$ is assumed to be a full-rank matrix, thus we have $\mbox{\boldmath $KK$}^+ = \mbox{\boldmath $KK$}^{-1} = \mbox{\boldmath $I$}$ which can be used to demonstrate the symmetry of the product $\mbox{\boldmath $G$}^\top\mbox{\boldmath $GKK$}^+$. Interestingly, Assumption B2 is specific to the BF test, while Assumptions B4 and B5 are in common with the $C(\alpha)$-type statistic and, are also required for Lagrange multiplier-type tests Dufour:2017. The set of conditions of Theorem (ref) allows us to introduce the following correction of the bilinear form statistic that is invariant to the definition of the null hypothesis.
Based on the simulation study by Lafontaine:1986, Goh:1996 examined the small-sample performance of corrections to the Wald statistic proposed by Phillips:1988. These corrections, based on Edgeworth expansions, aim to accelerate the convergence of the test statistic to its asymptotic distribution under two algebraically equivalent null hypotheses, namely:
where $k$ is a non-zero integer, and $\gamma$, $\beta$ are regression coefficients in the following model:
with $\mbox{\boldmath $x$}_1$, $\mbox{\boldmath $x$}_2$ denoting $n\times 1$ vector of covariates and $\mbox{\boldmath $\epsilon$}$ representing an $n\times 1$ vector of random disturbances.
Next, we examine the performance of the proposed correction for the BF test statistic by carrying out a comparison against the Wald $(W)$, Lagrange multiplier $(LM)$, bilinear form $(BF)$, and metric distance $(D)$ statistics considering the equivalent hypotheses given in (ref) and (ref). For definitions of each of these statistics in the context of extremum estimation refer to Dufour:2017, Crudu:2020 and Newey:1987, respectively.
Following the simulation study reported by Goh:1996, we considered model given in (ref) with $\mbox{\boldmath $\theta$} = (\gamma,\beta)^\top = (1,1)^\top$ and hypotheses defined in (ref) and (ref) for $k \in \{-5,-2,2,5\}$. $10\,000$ were generated with sample sizes $n=25, 50, 100$ and $500$ from model (ref) where $\mbox{\boldmath $x$}_j = (x_{1j},\dots,x_{nj})^\top$ with $x_{ij}\sim\mathsf{U}(0, 1)$ for $i=1,\dots,n$; $j=1,2$, and $\mbox{\boldmath $\epsilon$}\sim\mathsf{N}_n(\mbox{\boldmath $0$}, \sigma^2\mbox{\boldmath $I$})$, where $\sigma^2 = 1$.
For the hypotheses $H_0$ and $H_0^*$ defined in (ref) and (ref), respectively, we obtain that the reparameterization $\mbox{\boldmath $\phi$}(\mbox{\boldmath $\theta$}) = \mbox{\boldmath $\phi$}(\gamma,\beta)$, with \[ \mbox{\boldmath $\phi$}^{-1}(\gamma_*,\beta_*) = (\gamma_*, \beta_*^k)^\top, \] satisfies the condition B6. In accordance with hypothesis (ref) we obtain \[ \boldmath $K$(\boldmath $\theta$_*) =
, \qquad \boldmath $G$_*^+(\boldmath $\theta$_*) =
, \] and the verification of conditions B1 and B2 is straightforward. In this example we have that $\mbox{\boldmath $B$} = -\mbox{\boldmath $A$}$, which yields to a simplified version of the BF statistic. Thus, only conditions B1, B3 and B6 are required to obtain the corrected test statistic given in Equation (ref).
In the Monte Carlo experiment a $5\%$ level test was performed.\footnote{The replication files related to this article are available online at \url{https://github.com/faosorios/BF_invariance}} Table (ref) displays the results of testing $H_0$ and $H_0^*$ for four values of $k$. As expected all statistics are in agreement with the linear hypothesis in (ref), whereas the Wald statistic and the uncorrected version for the BF statistic show obvious invariance problems with empirical sizes that approach the nominal level as the sample size increases. The results suggest that the proposed correction (reported as $BF^c$ in Table (ref)) allows the empirical size of the BF statistic to be consistently closer to the nominal value, and except for some cases when the sample size is small, e.g. for $k = 2$ and $n = 25$. It is worth noting that this test aligns with the metric distance and Lagrange multiplier statistics, $D$ and $LM$ which are known to be invariant.
In this paper we study conditions that must be imposed on the BF statistic to avoid the undesirable property of non-invariance to reformulations of the hypothesis of interest generated by reparametrizations. Although the results of the simulation study seem promising, it should be noted that the required conditions are not always satisfied. For example, for the problem introduced by Gregory:1985 with two equivalent null hypotheses, \[ H_0: \theta_1 - 1/\theta_2 = 0, \qquad \textrm{and} \qquad H_0^* : \theta_{*1}\theta_{*2} - 1 = 0, \] we may note that the reparametrization $\mbox{\boldmath $\phi$}(\theta_1,\theta_2)$, with \[ \mbox{\boldmath $\phi$}^{-1}(\theta_{*1},\theta_{*2}) = ((\theta_{*1} + 1)\theta_{*2} - 1,1/\theta_{*2})^\top, \] satisfies the condition in (ref), but it does not satisfy Assumption B2. Thus, as with the Wald test, caution is required in the use of the BF statistic. The Monte Carlo experiment in Section (ref) is insightful, as it highlights the deterioration of the uncorrected BF statistic as nonlinearity increases. In light of the above results, further research is needed to better understand the properties and finite sample behavior of the BF statistic. A promising avenue of research is the application of the corrections based on the asymptotic expansions developed in Phillips:1988.
This work was written while the second author was at Universidad T\'ecnica Federico Santa Mar\'ia. The authors acknowledge the support of the computing infrastructure of the Applied Laboratory of Spatial Analysis UTFSM\,-\,ALSA (MT7018). \'Angelo G\'arate was supported by the National Scholarship Program of Chile, ANID-Chile. Federico Crudu's research benefited from financial support provided by the University of Siena via the F-CUR grant 2274-2022-CF-PSR2021-FCUR 001.