Unit Root Tests in Time Series and Stochastic Volatility Models

Abstract

Providing appropriate forecasts of time series data into the future depends crucially on whether the time series under consideration is non-stationary (i.e. has a unit root) or stationary. In the context of a Stochastic Volatility Model (SVM), the presence of a unit root in financial data has important implications for the pricing of various financial instruments. We propose a unit root test for the volatility process based on the Simulation-Extrapolation (SIMEX) approach. We express the SVM as a measurement error model and propose a Simulation-Extrapolation (SIMEX)-based approach to test for the unit root hypothesis. The asymptotic theory of the Ordinary Least Squares (OLS) and Weighted Symmetric (WS) estimators are exploited to obtain SIMEX-based tests and simulation studies are provided to demonstrate that the SIMEX-based test compares favorably with some of the well known unit root tests already available in the literature.

We also propose a unit root test based on the maximum order statistic in a simple autoregressive (AR) model of order 1. The asymptotic distribution of the test statistic under the null hypothesis is derived and the approximate percentiles are also provided. Through simulation studies, the proposed test is compared with the Dickey-Fuller (DF) test under various specifications for the error distributions.

In the final chapter of this dissertation, we propose a procedure to test the null hypothesis of stationarity in AR (1) models. The procedure is based on the Intersection-Union tests used in Bio-Equivalence studies. The performance of the test based on finite sample percentiles as well as asymptotic percentiles is assessed using simulation studies.