Testing for a unit root with nonstationary nonlinear heteroskedasticity

We provide a large sample theory for the Dickey-Fuller unit root test when the volatility process is driven by a nonlinear transformation of nonstationary time series. Our theory allows the dynamics of future volatilities being affected by the current shock, and involves replacing the nuisance nonlinear function by its consistent kernel estimator. This improves the existing literature for unit root testing with heteroskedasticity by using external data explicitly. We further propose a valid bootstrap procedure to implement the test, which is found to perform well in finite samples. A real data example is also provided