Rate-optimal data-driven specification testing in regression models

We propose a general procedure for testing that a regression function has a prescribed parametric form. We allow for multivariate regressors, non-normal errors and heteroscedasticity of unknown form. The test relies upon a nonparametric linear estimation method, such as a sieves expansion or the kernel method. The choice of the smoothing parameter is data-driven. Under the null hypothesis, the asymptotic distribution of the test statistic is the standard normal distribution. Use of bootstrap critical values is formally justified. The test is shown to be adaptive and rate-optimal in the minimax sense. Detection of Pitman-type local alternatives is also studied.