Bootstrap co-integration rank testing: the role of deterministic variables and initial values in the bootstrap recursion

In this paper we investigate the role of deterministic components and initial values in bootstrap likelihood ratio type tests of cointegration rank. A number of bootstrap procedures have been proposed in the recent literature some of which include estimated deterministic components and nonzero initial values in the bootstrap recursion while others do the opposite. To date, however, there has not been a study into the relative performance of these two alternative approaches. In this paper we fill this gap in the literature and consider the impact of these choices on both ordinary least squares (OLS) and generalized least squares (GLS) detrended tests, in the case of the latter proposing a new bootstrap algorithm as part of our analysis. Overall, for OLS detrended tests our findings suggest that it is preferable to take the computationally simpler approach of not including estimated deterministic components in the bootstrap recursion and setting the initial values of the bootstrap recursion to zero. For GLS detrended tests, we find that the approach of Trenkler (2009), who includes a restricted estimate of the deterministic component in the bootstrap recursion, can improve finite sample behavior further.