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 non-zero mean function. We exploit the recent work by Cho, Phillips, and Seo (2021) and show that the associated Wald test statistics have standard chi-square limiting null distributions, standard non-central 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, that involves nuisance parameters identified only under the alternative. In Monte Carlo studies, we find that our tests have well behaved levels. We also find that functional ordinary least squares tests can have power better than existing methods that do not exploit this covariance structure, like the specification testing procedures of Bierens (1982, 1990) or Stinchcombe and White(1998). Finally, we apply our methodology to the probit models for voting turnout that are estimated by Wolfinger and Resenstone (1980) and Nagler (1991) and test whether the models are correctly specified or not.