Testing the fractional integration parameter revisited: a fractional Dickey-Fuller test (Q2204611)

Summary: The main scope of this paper is to provide how to extend the standard Dickey-Fuller test [\textit{D. A. Dickey} and \textit{W. A. Fuller}, J. Am. Stat. Assoc. 74, 427--431 (1979; Zbl 0413.62075)] by taking into account the fractional case. Such extension has already been discussed by \textit{J. J. Dolado} et al. [Econometrica 70, No. 5, 1963--2006 (2002; Zbl 1132.91590)]. In this paper, we show, in the first step, that the fractional Dickey-Fuller test proposed by Dolado et al. is useless in practice. In the second step we show how to extend adequately the standard framework of Dickey-Fuller test to take into account the fractional case by using the usual test statistics and the usual asymptotic distributions [\textit{P. C. B. Phillips}, ibid. 55, 277--301 (1987; Zbl 0613.62109)]. Such extension can be very useful in practice. Through a simulation study, we show the good performance of the test in terms of size and power. Finally, in order to show how to use the new testing procedure, the test is applied to the well-known Nelson and Plosser data.