Fractional Dickey-Fuller tests under heteroskedasticity

In a recent paper, Dolado, Gonzalo and Mayoral (2002) introduce a fractional Dickey-Fuller (FD-F) t-statistic for testing a unit root against the alternative of a mean reverting fractional unit root process. This t-statistic is based on the assumption that the errors are unconditionally homoskedastic. However, Busetti and Taylor (2003), McConnell and Perez-Quiros (2000), and van Dijk et al. (2002) have found compelling evidence that such an assumption is unlikely to hold in many macroeconomic and …nancial time series, especially those obtained at a longer time span. In this paper, we investigate the …nite-sample properties of the FD-F statistic when the errors are unconditionally heteroskedastic. We …nd that, depending on the form of heteroskedasticity, the FD-F statistic su¤ers from substantial size distortions. In order to correct for such distortions, we propose the use of White standard errors (White (1980)) when computing the FD-F statistic. This yields a test that is robust to heteroskedasticity of unknown form. We demonstrate that the FD-F statistic that employs White standard error has a standard normal limiting distribution under the unit root null hypothesis as in the FD-F statistic with homoskedastic errors. Monte Carlo results suggest that: (i) White’s correction is e¤ective in reducing the size distortions; and (ii) the power loss of using White standard error in the case of homoskedasticity is very small. These results suggest that it is prudent to use the White robust standard errors regardless of whether the errors are heteroskedastic or not. Keywords: Fractional unit root tests, heteroskedasticity, structural breaks J.E.L. Classi…cation: C30, C32 1 Introduction Several articles have provided Monte Carlo evidence on the performance of the unit-root tests, stationary tests and persistence change tests under unconditional heteroskedasticity. The general …nding from these articles is that these tests su¤er from severe size distortions when there is an abrupt change in the error variance. In standard unit root testing against stationarity alternatives, Hamori and Tokihisa (1997) study the Dickey-Fuller (D-F) test, Cavaliere and Taylor (2007) study theM test of Stock (1999) and Beare (2007) investigates the Phillips-Perron test. In the stationarity testing framework, where the null hypothesis