The primary focus of this article is the provision of tests for additional conditional moment constraints in cross-section or short panel data contexts. The principal contribution is the explicit incorporation of conditional moment restrictions defining the maintained hypothesis in the formulation of the test statistics thus mirroring that of the classical parametric likelihood setting by defining restricted tests in contradistinction to unrestricted tests that partially or completely ignore the maintained moment condition information. The framework is quite general allowing the parameters defining the additional and maintained conditional moment restrictions to differ and permitting the conditioning variates to differ likewise. GMM and generalized empirical likelihood test statistics are suggested. The asymptotic properties of the statistics are described under both null hypothesis and a suitable sequence of local alternatives. An extensive set of simulation experiments explores the practical efficacy of the various test statistics in terms of empirical size and size-adjusted power confirming the superiority of restricted over unrestricted tests. A number of restricted tests possess both sufficiently satisfactory empirical size and power characteristics to allow their recommendation for econometric practice.