Testing a Class of Semi- or Nonparametric Conditional Moment Restriction Models using Series Methods

This paper proposes a new test for a class of conditional moment restrictions whose parameterization involves unknown, unrestricted conditional expectation functions. Examples of such conditional moment restrictions are conditional mean independence (leading to a nonparametric significance test) and conditional homoskedasticity (with an otherwise unrestricted conditional mean) and also arise from models of single-agent discrete choice under uncertainty and static games of incomplete information. The proposed test may be viewed as a semi-/nonparametric extension of the Bierens (1982) goodness-of-fit test of a parametric model for the conditional mean. Estimating conditional expectations using series methods and employing a Gaussian multiplier bootstrap to obtain critical values, the resulting test is shown to be asymptotically correctly sized and consistent. A simulation study applies the procedure to test the specification of a two-player, binary-action static game of incomplete information, treating equilibrium beliefs as nonparametric conditional expectations.