Sequentially testing polynomial model hypotheses using power transforms of regressors

We provide a methodology for testing a polynomial model hypothesis by generalizing the approach and results of Baek, Cho, and Phillips (Journal of Econometrics, 2015, 187, 376–384; BCP), which test for neglected nonlinearity using power transforms of regressors against arbitrary nonlinearity. We use the BCP quasi-likelihood ratio test and deal with the new multifold identification problem that arises under the null of the polynomial model. The approach leads to convenient asymptotic theory for inference, has omnibus power against general nonlinear alternatives, and allows estimation of an unknown polynomial degree in a model by way of sequential testing, a technique that is useful in the application of sieve approximations. Simulations show good performance in the sequential test procedure in both identifying and estimating unknown polynomial order. The approach, which can be used empirically to test for misspecification, is applied to a Mincer (Journal of Political Economy, 1958, 66, 281–302; Schooling, Experience and Earnings, Columbia University Press, 1974) equation using data from Card (in Christofides, Grant, and Swidinsky (Eds.), Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp, University of Toronto Press, 1995, 201-222) and Bierens and Ginther (Empirical Economics, 2001, 26, 307–324). The results confirm that the standard Mincer log earnings equation is readily shown to be misspecified. The applications consider different datasets and examine the impact of nonlinear effects of experience and schooling on earnings, allowing for flexibility in the respective polynomial representations

Sequentially Testing Polynomial Model Hypotheses Using Power Transforms of Regressors

We provide a methodology for testing a polynomial model hypothesis by extending the approach and results of Baek, Cho, and Phillips (2015; Journal of Econometrics; BCP) that tests for neglected nonlinearity using power transforms of regressors against arbitrary nonlinearity. We examine and generalize the BCP quasi-likelihood ratio test dealing with the multifold identification problem that arises under the null of the polynomial model. The approach leads to convenient asymptotic theory for inference, has omnibus power against general nonlinear alternatives, and allows estimation of an unknown polynomial degree in a model by way of sequential testing, a technique that is useful in the application of sieve approximations. Simulations show good performance in the sequential test procedure in identifying and estimating unknown polynomial order. The approach, which can be used empirically to test for misspecification, is applied to a Mincer (1958, 1974) equation using data from Card (1995). The results confirm that Mincer’s log earnings equation is easily shown to be misspecified by including nonlinear effects of experience and schooling on earnings, with some flexibility required in the respective polynomial degrees

Testing Linearity Using Power Transforms of Regressors

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