  This paper proposes a new class of nonparametric tests for the correct
specification of models based on conditional moment restrictions, paying
particular attention to generalized propensity score models. The test procedure
is based on two different projection arguments, leading to test statistics that
are suitable to setups with many covariates, and are (asymptotically) invariant
to the estimation method used to estimate the nuisance parameters. We show that
our proposed tests are able to detect a broad class of local alternatives
converging to the null at the usual parametric rate and illustrate its
attractive power properties via simulations. We also extend our proposal to
test parametric or semiparametric single-index-type models.
