Testing for neglected nonlinearity in regression models: A collection of new tests based on the theory of random ...elds¤

Within the framework of the ‡exible regression model approach proposed by Hamilton (2000), we demonstrate the use of the theory of random ...elds as a building block for a battery of new tests for neglected nonlinearity. The common denominator to all the tests is that they are TR versions of the Lagrange multiplier statistic. The proposed tests are free of unidenti...ed nuisance parameters, contrary to Hamilton’s and the popular Neural Network tests, where the number of unidenti...ed nuisance parameters under the null hypothesis increases with the dimensionality of the model. Furthermore, the proposed tests are robust to the speci...cation of the variance-covariance function of the random ...eld, and their computation is simple compared to the statistic proposed by Hamilton, since they are based on auxiliary regressions. These advantages are the result of 1) studying alternative representations of the random ...eld governing the outcome of the conditional mean in Hamilton’s regression model, and 2) using local approximations to the variance-covariance function based on Taylor’s expansion. The main modi...cation to the random ...eld consists of switching from the Euclidean distance measure to the Minkowski distance measure. A Monte Carlo simulation study suggests that our statistic based on the Minkowski distance broadly matches the power properties of the statistic suggested by Hamilton. The tests match the simplicity of the ¤The paper has bene...tted from numerous discussions and comments from Peter Storgaard. We would also like to thank Niels Haldrup, Svend Hylleberg, Alvaro Escribano, Esben Høg, Peter Jensen and Allan Würtz and seminar participants at the University of Aarhus, University of California, Riverside, and the Econometric Society Meetings in Boston. The second author acknowledges the ...nancial support of the Academic Senate of the University of California, Riverside.