Alternative Bootstrap Procedures for Testing Cointegration in Fractionally Integrated Processes

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The special feature of the fractional case is the dependence of the asymptotic null distributions of conventional statistics on the fractional integration parameter. Such tests are said to be asymptotically non-pivotal, and conventional asymptotic tests are therefore not available. Bootstrap tests can be constructed, although these may be less reliable in small samples than in the case of asymptotically pivotal statistics. Bias correction techniques, including the double bootstrap of Beran (1988) and the fast double bootstrap of Davidson and MacKinnon (2000) are considered. The investigation focuses on the issues of (a) choice of statistic, (b) bias correction techniques, and also (c) designing the simulation of the null hypothesis. The latter consideration is crucial for ensuring tests are both correctly sized and powerful. Three types of test are considered that, being envisaged as routine exploratory tools, are based on the residuals from a putative cointegrating regression. Two are of the null hypothesis of non-cointegration; a conventional residual-based test using the Durbin-Watson statistic, and a test based on the F -statistic, as proposed in Davidson (2002). The third is the Shin (1994) residual-based test of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments whose main object is to throw light on the relative roles of issues (a), (b) and (c) in test performance.