In this paper we analyze the consequences of model overidentification on testing exogeneity, when maximum likelihood technique for estimation an inference are used. This situation is viewed as a particular case of the more general problem of considering how restrictions on "nuisance" parameters could help in making inference on the parameters of interest. At first general model is considered. A suitable likelihood function factorization is used which permits to easily derive the information matrix and other tools useful for constructing hoint tests for exogeneity and overidentifying restrictions both of Wald and Lagrange Multiplier type. The asymptotic local power of the exogeneity test in the justidentified model is compared with that of the overidentified one, when we assume that this last is the true model. Then the pseudolikelihood framework is proposed for appreciating the consequences of working woth a model where overidentifying restrictions are erroneously imposed. The inconsistency introduced by imposing false restrictions is analysed and the consequences of the misspecification on the exogeneity test are carefully examined.
