Distribution theory for unit root tests with conditional heteroskedasticity

This paper explores the asymptotic distribution theory of autoregressive (AR) unit root tests where the error follows a generalized autoregressive conditional heteroskedastic (GARCH) process. The proposed unit root test is based on maximum likelihood estimation, which estimates the AR unit root and the GARCH parameters jointly. The asymptotic distribution of the t-statistic for the AR unit root is a mixture of the Dickey-Fuller t-distribution and the standard normal, with the relative weight depending on the magnitude of the GARCH effect and the fourth moment of the standardized errors. As the GARCH effect increases, the power of the tests improves significantly. These results show that significant power gains emerge from the joint estimation rather than relying on the conventional ADF test which ignores the heteroskedasticity in the data.