Specification testing with locally misspecified alternatives

It is well known that most of the standard specification tests are not robust when the alternative is misspecified. Using the asymptotic distributions of standard Lagrange multiplier (LM) test under local misspecification, we suggest a robust specification test. This test essentially adjusts the mean and covariance matrix of the usual LM statistic. We show that for local misspecification the adjusted test is asymptotically equivalent to Neyman's C(α) test, and therefore, shares the optimality properties of the C(α) test. The main advantage of the new test is that, compared to the C(α) test, it is much simpler to compute. Our procedure does require full specification of the model and there might be some loss of asymptotic power relative to the unadjusted test if the model is indeed correctly specified.