Alpha-Stable Consistent Model Specification Tests for Heavy-Tailed Neural Networks Environments

This paper investigates applications of stable-law limiting theory to model specification tests in which non-linearities are sought in data that exhibit bounded maximal moments. Utilizing the stable-laws allows us for the first time to prove that consistent conditional moment tests (CM) of a functional form within neural network environments are not chi-squared in distribution. In addition, we prove that CM tests suffer a dramatic loss in power when moments greater than two are infinite. Furthermore, we offer for the first time a set of computationally cheapest statistics that are stable-functionals of suitable moment conditions. The new statistics are suitable for all iid and serially dependent data processes and are directly applicable to neural network learning in financial time-series models. The stable-law statistics are invariant to moment condition failure, remain maximally powerful under mild conditions, and do not require a restrictive orthogonality condition under the null hypothesis. Simulation experiments indicate that CM tests are far more likely to predict non-linearity erroneously in data than true chi-squared distributions imply. Moreover, in comparison, for certain data environments, the new stable-law statistics demonstrate perfect power for all levels of moment condition failure.