Linearity testing using local polynomial approximation

We use local polynomial approximation to estimate the conditional mean and conditional variance, and test linearity by using a functional measuring the deviation between the nonparametric estimates and the parametric estimates based on a linear model. We also employ first- and second-order derivatives for this purpose, and we point out some advantages of using local polynomial approximation as opposed to kernel estimation in the context of linearity testing. The asymptotic theory of the test functionals is developed in some detail for a special case. It is used to draw qualitative conclusions concerning the bandwidth, but in order to apply the asymptotic distribution to specific testing problems very large sample sizes are needed. For moderate sample sizes we have examined a bootstrap alternative in a large variety of situations. We have tried bandwidths suggested by asymptotic results as well as bandwidths obtained by cross-validation.