Parametric and Semiparametric Efficient Tests for Parameter Instability

This paper proposes asymptotically point optimal tests for parameter instability under the feasible circumstance that the researcher has little information about the unstable parameter process and the error distribution. The shape of the unstable parameter process is not identified but is asymptotically described by the Winer process, which is weak enough to cover a wide range of structural breaks and time varying parameter processes. I first derive a test under known error distribution, and show that the test is asymptotically equivalent to likelihood ratio tests for correctly identified unstable parameter processes under suitable conditions. The test is then extended to semiparametric models in which the underlying distribution is unknown but treated as an infinite dimensional nuisance parameter. An adaptive test is shown to be attainable without further restrictive conditions on the error distribution, which implies that the semiparametric power envelope is asymptotically equivalent to that of parametric models.