In earlier chapters we have considered at some length the consequences of correct specification and misspecification. In this chapter, we consider statistical methods for detecting the presence of misspecification.

Specific methods for detecting misspecification are based on the contrasting consequences of correct specification and misspecification. For example, when a model is correctly specified there are usually numerous consistent estimators for the parameters of interest (e.g., ordinary least squares and weighted least squares). If the model is correctly specified, these different estimators should have similar values asymptotically. If these values are not sufficiently similar, then the model is not correctly specified. This reasoning forms the basis for the Hausman [1978] test, a special case of which we encountered in the previous chapter. Such tests have power because of the divergence of alternative estimators under misspecification. As another example, correct specification implies the validity of the information matrix equality. If estimators for -A*n and B*n are not sufficiently similar, then one has empirical evidence against the validity of the information matrix equality and thus against the correctness of the model specification. This reasoning forms the basis for the information matrix tests (White [1982, 1987]). Such tests have power because of the failure of the information matrix equality under misspecification.