We propose a Vuong-type model-selection test for models defined by conditional moment restrictions. The moment restrictions that define the models can be standard equality restrictions that point-identify the model parameters, or moment equality or inequality restrictions that partially identify the model parameters. The test uses a new average generalized empirical likelihood criterion function designed to incorporate full restriction of the conditional model. We also introduce a new adjustment to the test statistic that makes it asymptotically pivotal whether the candidate models are nested or non-nested. The test uses simple standard normal critical values and is shown to be asymptotically similar, to be consistent against all fixed alternatives, and to have non-trivial power against -local alternatives. Monte Carlo simulations demonstrate that the finite sample performance of the test is in accordance with the theoretical prediction.