Fractional Dickey-Fuller test with or without prehistorical influence

Recently, the generalization of the standard Dickey-Fuller test to the fractional case has been proposed. This test, called the fractional Dickey-Fuller (<i>F</i><i>D</i><i>F</i>) test, is built on the basis of a composite null hypothesis. The <i>F</i><i>D</i><i>F</i> test can be applied to a sample generated from a type <i>I</i> or a type <i>I</i><i>I</i> fractional process. Depending on whether the test is applied to a sample generated from a type <i>I</i> or type <i>I</i><i>I</i> process, it is referred to as a test with or without prehistoric influence, respectively. The primary goal of this research is to investigate the effect of a pre-sample on the finite sample null distribution. The second objective is to highlight the theoretical justifications for the choice of the null composite hypothesis. All the theoretical results are illustrated with simulated and real data sets. Furthermore, to facilitate the reproducibility of our simulation data and figures, we provide all the necessary supplementary material consisting of EViews programs.