Goodness-of-fit tests for kernel regression with an application to option implied volatilities

This paper proposes a test of a restricted specification of regression, based on comparing residual sum of squares from kernel regression. Our main case is where both the restricted specification and the general model are nonparametric, with our test equivalently viewed as a test of dimension reduction. We discuss practical features of implementing the test, and variations applicable to testing parametric models as the null hypothesis, or semiparametric models that depend on a finite parameter vector as well as unknown functions. We apply our testing procedure to option prices; we reject a parametric version of the Black–Scholes formula but fail to reject a semiparametric version against a general nonparametric regression.