We develop a method of testing linearity using power transforms of regressors, allowing for stationary processes and time trends. The linear model is a simplifying hypothesis that derives from the power transform model in three diﬀerent ways, each producing its own identiﬁcation problem. We call this modeling diﬀiculty the trifold identiﬁcation problem and show that it may be overcome using a test based on the quasi-likelihood ratio (QLR) statistic. More speciﬁcally, the QLR statistic may be approximated under each identiﬁcation problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis. The limit theory for the QLR test statistic depends on a Gaussian stochastic process. In the important special case of a linear time trend regressor and martingale diﬀerence errors asymptotic critical values of the test are provided. The paper also considers generalizations of the Box-Cox transformation, which are associated with the QLR test statistic.
Baek, Yae In; Cho, Jin Seo; and Phillips, Peter C.B., "Testing Linearity Using Power Transforms of Regressors" (2013). Cowles Foundation Discussion Papers. 2305.