Specification Test based on Convolution-type Distribution Function Estimates for Non-linear Auto-regressive Processes

The paper proposes a specification test based on two estimates of distribution function. One is the traditional kernel distribution function estimate and the other is a newly proposed convolution-type distribution function estimate. Asymptotic properties of the new estimate are studied when the innovation density is known and when it is unknown. The MISE-type statistic based on these estimates is suggested to test parametric specifications of the mean and volatility functions. The relating asymptotic results are obtained and the finite-sample properties are studied based on the bootstrap methodology. A simulation study shows that the proposed test competes favorably to benchmark tests in terms of the empirical level and power.