Goodness-of-fit tests for functional data

Economic data are frequently generated by stochastic processes that can be modelled as occurring in continuous time. That is, the data are treated as realizations of a random function (functional data). Sometimes an economic theory model specifies the process up to a finite‐dimensional parameter. This paper develops a test of the null hypothesis that a given functional data set was generated by a specified parametric model of a continuous‐time process. The alternative hypothesis is non‐parametric. A random function is a form of infinite‐dimensional random variable, and the test presented here a generalization of the familiar Cramér‐von Mises test to an infinite dimensional random variable. The test is illustrated by using it to test the hypothesis that a sample of wage paths was generated by a certain equilibrium job search model. Simulation studies show that the test has good finite‐sample performance.