Testing the Adequacy of a Linear Model via Critical Smoothing

An important problem for tting local linear regression is the choice of the smoothing parameter. As the smoothing parameter becomes large, the estimator tends to a straight line, which is the least squares t in the ordinary linear regression setting. This property may be used to assess the adequacy of a simple linear model. Motivated by Silverman's (1981) work for testing multimodality in kernel density estimation, a suitable test statistic for checking linearity is the critical smoothing parameter where the estimate changes from nonlinear to linear, determined by the approximate F-tests given in Hastie and Tibshirani (1990) for a prescribed type I error. To assess the significance , the \wild bootstrap" procedure is used to replicate the data and the proportion of bootstrap samples which give a nonlinear estimate when using the critical bandwidth is obtained as the p-value. Simulation results show that the critical smoothing test is useful in detecting a wide range of alternatives.