Bootstrap tests for fractional integration and cointegration: A comparison study

There are few methods in the literature to test for integration and cointegration in the traditional framework, i.e. using the I(0)–I(1) paradigm. In the first case, the most known are the Dickey–Fuller (DF), the Augmented Dickey–Fuller (ADF) and the Phillips–Perron (PP) tests, while in the latter case, the Engle and Granger (EG) and Johansen procedures are broadly used. But how well do these methods perform when the underlying process presents the long-memory characteristic? The bootstrap technique is used here to approximate the distribution of integration and cointegration test statistics based on a semiparametric estimator of the fractional parameter of ARFIMA(p,d,q) models. The proposed bootstrap tests, along with the asymptotic test based on the fractional semiparametric estimator, are empirically compared to the standard tests, for testing integration and cointegration in the long-memory context. Monte Carlo simulations are performed to evaluate the size and power of the tests. The results show that the conventional tests, except for the procedures based on the DF approach, loose power when compared to fractional tests. As an illustration, the tests were applied to the series of Ibovespa (Brazil) and Dow Jones (USA) indexes and led to the conclusion that these series do not share a long-run relationship.