Revisiting Tests for Neglected Nonlinearity Using Artificial Neural Networks

Abstract

Tests for regression neglected nonlinearity based on artificial neural networks (ANNs) have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold. This implies that the asymptotic behavior of general ANN-based tests for neglected nonlinearity is still an open question. Here we analyze a convenient ANN-based quasi-likelihood ratio statistic for testing neglected nonlinearity, paying careful attention to both components of the null. We derive the asymptotic null distribution under each component separately and analyze their interaction. Somewhat remarkably, it turns out that the previously known asymptotic null distribution for the type 1 case still applies, but under somewhat stronger conditions than previously recognized. We present Monte Carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new, stronger regularity conditions are violated.

Higher-Order Approximations for Testing Neglected Nonlinearity

“…We begin by specifying the same data-generating process (DGP) assumed by Cho et al (2011): Assumption 1 (DGP). Let ( , F , P) be a complete probability space on which is defined the strictly stationary and absolutely regular process {(Y t , X t ) ∈ R 1+k : t = 1, 2, .…”

“…As Cho et al (2011) show, different orders of expansion are required when testing λ * = 0 than when testing δ * = 0. A quadratic expansion is sufficient for testing λ * = 0 when δ = 0 (Hansen, 1996), whereas a quartic approximation is needed for testing δ * = 0, under regularity conditions provided by Cho et al (2011).…”

“…Recently, Cho, Ishida, and White (2011) showed that QLR tests for neglected nonlinearity based on artificial neural networks (ANNs) cannot be analyzed using quadratic approximation, and they provide conditions under which a quartic (fourth-order) approximation yields the desired firstorder asymptotics. Nevertheless, they also discuss the fact that cases violating their assumption A7 ("no zero") require the use of even higher-order approximations to obtain the first-order asymptotics for the QLR statistic.…”

“…The goal of this study is to gain a deeper understanding of the asymptotic behavior of ANN-based QLR tests for neglected nonlinearity when Cho et al's (2011) no-zero assumption is violated. In doing so, we illustrate the use of higher-order, specifically sixth-order, expansions to obtain first-order asymptotics.…”

“…In doing so, we illustrate the use of higher-order, specifically sixth-order, expansions to obtain first-order asymptotics. Although Cho et al (2011) obtain the asymptotic distribution of their QLR statistic by explicitly treating the twofold identification problem that arises in this approach to testing neglected nonlinearity, for conciseness we restrict our focus here to analyzing the QLR statistic when there is only a single source of identification failure under the null. We leave the twofold identification problem to other work.…”