OPTIMALITY FOR THE INTEGRATED CONDITIONAL MOMENT TEST

Abstract

This paper proposes a version of the integrated conditional moment (ICM) test that is optimal for a class of composite alternatives. The ICM test is built on the fact that a random function based on a correctly specified model should have zero mean, whereas any misspecification in the conditional mean implies a divergent mean for the random function. We derive test statistics that are optimal for each basis element of an orthonormal decomposition of the function space for which the random function is an element. We then use a weighted summation of these test statistics to compose the single test statistic that is optimal for any pair of alternatives that are symmetric about zero. This test is equivalent to using a particular measure in the ICM test of Bierens and Ploberger (1997, Econometrica 65, 1129–1152).