On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qLL test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the sup-Wald test does not show this non-monotonicity and its power is far superior to the qLL test when the break is not very small. We claim that the optimality of the qLL test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the sup-Wald test indeed dominates the qLL test and, in many cases, the latter has zero relative asymptotic efficiency.