Chi-square Tests for Parameter Stability

Testing when a nuisance parameter is identified only under the alternative is problematic because the Likelihood Ratio test converges to a nonstandard distribution that may depend on unknown parameters. Examples include testing parameter stability in Structural Change and Threshold models. Our article proposes a class of tests that have the advantage of having a standard distribution, namely a chi-square. In this class, we focus mostly on a Lagrange Multiplier test in an auxiliary regression. We derive the asymptotic power of this test against alternatives which differ from the implicit alternative of the test. We show that this test can be used as a diagonistic test for parameter stability. A Monte Carlo study compares the performance of our tests with other frequently used tests and shows that they have a similar power.