We consider testing distributional assumptions by using moment conditions. A general class of moment conditions satisfied under the null hypothesis is derived and connected to existing moment-based tests. The approach is simple and easy to implement, yet reasonably powerful. In addition, we provide moment tests that are robust against parameter estimation error uncertainty in the general case which includes the case of serial correlation. In particular, we consider the location-scale model for which we derive robust moment tests, regardless of the forms of the conditional mean and variance. We study in detail the Student and inverse Gaussian distributions. Simulation experiments are conducted to assess the finite sample properties of the tests. We provide two empirical examples on foreign exchange rates by testing the Student distributional assumption of T-GARCH daily returns and on daily realized variance by testing the inverse Gaussian distributional assumption.
