Abstract
An important problem for fitting 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 fit 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 in kernel density estimation, a suitable test statistic is the critical smoothing parameter where the estimate changes from nonlinear to linear, while linearity or nonlinearity requires a more precise judgment. We define the critical smoothing parameter through the approximate F-tests by Hastie and Tibshirani (1990). 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.