Testing correct model specification using extreme learning machines

Testing the correct model specification hypothesis for artificial neural network (ANN) models of the conditional mean is not standard. The traditional Wald, Lagrange multiplier, and quasi-likelihood ratio statistics weakly converge to functions of Gaussian processes, rather than to convenient chi-squared distributions. Also, their large sample null distributions are problem dependent, limiting applicability. We overcome this challenge by applying functional regression methods of Cho, Huang, and White (2008) to extreme learning machines (ELM). The Wald ELM (WELM) test statistic proposed here is easy to compute and has a large-sample standard chi-squared distribution under the null hypothesis of correct specification. We provide associated theory for time-series data and affirm our theory with some Monte Carlo experiments.