Nonparametric Specification Testing for Hammerstein Systems

This paper examines the problem of testing a parametric form of the Hammerstein cascade nonlinear system against a nonparametric alternative. The paper proposes two test statistics relying on the orthogonal Hermite series nonparametric regression function estimate. An asymptotic theory of the proposed tests is established including the asymptotic normality under the null hypothesis that the parametric form of the Hammerstein system nonlinearity is correct. Moreover, under the general nonparametric alternative the probability of detecting that parametric model is incorrect tends to one. The finite sample performance of the established tests is also examined.