Kernel Test for Neglected Nonlinearity

摘要

Recent tests of the nonlinearity of the functional form a regression model have focused on attempts to develop consistent tests that do not specify a parametric alternative and encompassing forms. These tests are a special case of the broader conditional moment tests for misspecification, such as that of Bierens (1990), that have greater power by restricting the specification to the linear form. Examples of this are the tests of Lee, White and Granger (1993) (LWG), which is based on the Bierens test and which detects omitted nonlinearity through the use of a 'neural net', and Wooldridge (1992), which is based on the Davidson-Mackinnon test and uses a Fourier expansion. Like the test of Bierens, these tests are consistent against deviations from the null. In this paper we adapt the consistent test of Bradley and McClelland (1994) to a test of nonlinearity in a manner analogous to the adaptation of the Bierens test by LWG. Our test is based on the idea of Newey (1985) that if the model is not properly specified then the vector of explanatory variables, x, can better predict the residuals from a linear regression than the residual's mean. Thus, while many x-measurable functions, such as a neural net, can be used to detect neglected nonlinearities an appealing function to use is the conditional expectation of the residuals given x.