Testing a constant mean function using functional regression

In this paper, we study functional ordinary least squares estimator and its properties in testing the hypothesis of a constant zero mean function or an unknown constant nonzero mean function. We exploit the recent work by Cho, Phillips, and Seo (Int Econ Rev 170:391–456, 2022) and show that the associated Wald test statistics have standard chi-square limiting null distributions, standard noncentral chi-square distributions for local alternatives converging to zero at a n rate, and are consistent against global alternatives. These properties permit computationally convenient tests of hypotheses involving nuisance parameters. In particular, we develop new alternatives to tests for regression misspecification using the neural network model, that involves nuisance parameters identified only under the alternative. Our Monte Carlo simulations affirm the theory of the current study. Finally, we apply our methodology to the probit models for voter turnout that are estimated by Wolfinger and Rosenstone (Who votes? Yale University Press, New Haven, 1980), Nagler (Am Political Sci Rev 85:1393–1405, 1991) and test whether the models are correctly specified or not.