# Asymptotic Theory of Integrated Conditional Moment Tests

https://doi.org/0012-9682(199709)65:5<1129:ATOICM>2.0.CO;2-P
p. 1129-1151

Herman J. Bierens, Werner Ploberger

In this paper we derive the asymptotic distribution of the test statistic of a generalized version of the integrated conditional moment (ICM) test of Bierens (1982, 1984), under a class of $\sqrt n$-local alternatives, where $n$ is the sample size. The generalized version involved includes neural network tests as a special case, and allows for testing misspecification of dynamic models. It appears that the ICM test has nontrivial local power. Moreover, for a class of "large" local alternatives the consistent ICM test is more powerful than the parametric $t$ test in a neighborhood of the parametric alternative involved. Furthermore, under the assumption of normal errors the ICM test is asymptotically admissible, in the sense that there does not exist a test that is uniformly more powerful. The asymptotic size of the test is case-dependent: the critical values of the test depend on the data-generating process. In this paper we derive case-independent upperbounds of the critical values.