Goodness of Fit and Change Point Test for ACD Models and Explosive Period Detection of Financial Time Series

Abstract

In this thesis, we consider the statistical inference for autoregressive conditional duration(ACD) models and mildly explosive autoregressive model. First, we study the entropy test for the goodness of fit test in nonlinear ACD models. To implement a test, we explore the null limiting distribution of the residual empirical process from ACD models and verify that it has an asymptotic expansion form that consists of the true empirical process and extra terms yielded by parameter estimation. Then, we show that under regularity conditions, the proposed entropy test approximately follows a distribution that is free from the parameter estimation. We also illustrate a simulation study and a real data analysis. Second, we consider the parameter change test in nonlinear ACD models. Particularly, we use the cumulative sum test based on parameter estimates and verify that its limiting null distribution is the supremum of a Brownian bridge. A simulation study and real data analysis are provided for illustration. Finally, the limit distribution of the least squares estimator for mildly explosive autoregressive models with strong mixing innovations is established, which is shown to be Cauchy as in the iid case. The result is applied to identify the onset and the end of an explosive period of an econometric time series. Simulation and data analysis are also conducted to demonstrate the usefulness of the result.