Specification testing with estimated variables

This article proposes specification tests for economic models defined through conditional moments restrictions in which conditioning variables are estimated. There are two main motivations for this situation. First, the case when the conditioning variables are not directly observable, such as economic models, where innovations or latent variables appear as explanatory variables. Second, the case when the set of conditioning variables is too large to derive powerful tests, and hence, the original conditioning set is replaced by a constructed variable that is regarded as a good summary of it. We establish the asymptotic properties of the proposed tests, examine its finite sample behavior, and apply them to different econometric contexts. In some cases, the proposed approach leads to relevant tests that generalize well known specification tests, such as Ramsey's RESET test.