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.
