Bootstrapping the order selection test

Abstract

We consider bootstrap versions of the order selection test of Eubank and Hart and Kuchibhatla and Hart for testing lack-of-fit of regression models. For homoscedastic data, conditions are established under which the bootstrap level error is smaller (asymptotically) than that of the large sample test A new statistic is proposed to deal with the case of heteroscedastic data. The limiting distribution of this test statistic is derived and shown to depend on the unknown error variance function. This dependency makes using the large sample test a formidable task in practice. An alternative approximation is to apply bootstrap procedures. We propose various bootstrap tests, including ones based on the wild bootstrap. Simulation studies indicate that the wild bootstrap generally has good level and power properties, although sometimes power can be increased by appropriate smoothing of squared residuals. A real-data example is also considered to further illustrate the methodology.