Model specification tests with misspecified alternatives: Some robust and simultaneous approaches

It is well known that most of the standard specification tests are not robust when the alternative is misspecified. We consider the three types of typical misspecification encountered in econometric model specification testing, namely, complete misspecification, underspecification, and overspecification. In the case of complete misspecification the distribution under the alternative hypothesis does not belong to the data generating process (DGP), while underspecification refers to the alternative being a subset of a more general model representing the DGP. Overspecification is the case when the alternative hypothesis is overstated. Most likely, the first two types of misspecification are common in one-directional testing situation whereas the last one happens when multi-directional joint tests are applied based on an overparametrized alternative model. Following Haavelmo's work, we provide a simple example to illustrate the effects of misspecification on testing economic hypothesis. Then we find the asymptotic distributions of standard one-directional and multi-directional Lagrange multiplier (LM) tests under these three kinds of misspecification. The asymptotic relative efficiency (ARE) of the misspecified tests are also evaluated. Next using these distributions, we suggest a robust specification test under misspecified alternatives. The new test is shown to be asymptotically equivalent to Neyman's $C(\alpha)$ test. Some applications are presented to illustrate our theoretical results.