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A Kernel Test for Neglected Nonlinearity

Ralph Bradley and Robert McClelland


This paper develops a new kernel test for neglected nonlinearity in the conditional expectation function, and compares this test to the Ramsey RESET test (1969) and the Neural Net test of Lee, White, and Granger (1993). Like the Neural Test and the Ramsey Reset Test, this Kernel test is a Lagrange Multiplier test based on the R-Square statistic from a regression of the estimated residuals on the regressors, and an additional nonlinear function of the regressors. Unlike the other tests, our test has a two stage approach where in the first stage we estimate the structure of the misspecification and in the second stage we test for whether or not the estimate of the misspecification can better predict the residuals than their mean. This two stage approach can give the researcher guidance on the nature of the misspecification, and should improve the power of the test since the added function in the regression of the residuals is itself an estimate of the conditional expectation of the residuals given the independent variables. In addition, because it uses simple, well known estimation methods it can be easily implemented by researchers when using linear models.

Our test uses a cross-validated kernel regression estimate of the conditional mean of the residuals given the explanatory variables, and from the results of the Rao-Blackwell Theorem this should asymptotically be the nonlinear function of these variables whose correlation with the estimated residuals has the highest absolute value. As a result our test should have greater power at detecting nonlinearity than either the RESET or the Neural Net tests, even though we use an established estimation technique in a standard Lagrange Multiplier framework.

Because the kernel regression is an estimate of the conditional mean of the residuals, it contains information about the form of the nonlinearity. For example a research may plot the confidence interval and estimated conditional mean of the residuals, noting where and how the estimated mean diverges significantly from zero. One may also calculate the derivatives of the estimate at the mean for an indication of the direction of the misspecification.