Essays on optimal tests for parameter instability

There are a large number of tests for parameter instability designed for specific types of unstable parameter processes and error distributions. However, it is difficult to identify those types in practice based on a priori knowledge. My dissertation studies methods and conditions under which asymptotically efficient tests are obtained without the knowledge of the unstable parameter process and the error distribution. First, I examine asymptotically optimal tests for parameter instability in which the difficulty in identifying the unstable process is explicitly considered. Elliott and Muller (2006) provide conditions under which a large class of breaking processes lead to asymptotically equivalent optimal tests. Their finding, however, is restricted to linear Gaussian models. I improve upon their work in two ways. First, I show that the asymptotic equivalency of the efficient tests for parameter instability holds even in a broader set of parametric models which includes nonlinear models with non-Gaussian error distributions. It implies that the knowledge of the unstable parameter process is asymptotically irrelevant for testing purposes. Second, I suggest a test statistic that is asymptotically optimal for a broad set of unstable parameter processes. Second, I study asymptotically efficient tests for parameter instability in general semiparametric models in which the error distribution is unknown but treated as an infinite dimensional nuisance parameter. I first derive the asymptotic power envelope with unknown density and suggest conditions under which a semiparametric model would have the same asymptotic power envelope with known error distribution. The conditions are weak enough to cover a wide range of error distributions by relaxing the twice differentiability and allowing for skewness. An efficient test statistic is then suggested, which is adaptive in the sense that allowing unknown error distribution gives no loss of asymptotic power. This implies that the knowledge of the error distribution is asymptotically irrelevant under mild conditions. Finally, the suggested parameter instability tests are applied to various quantile models for U.S. inflation process such as Phillips curve, P- star model, and AR models. The tests result shows a strong evidence of parameter instability in most quantile levels of all model