Distribution-Free Tests Of Fractional Cointegration

We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series. Our test statistics have an asymptotic chi-square distribution under the null and rely on generalized least squares–type of corrections that control for the short-run correlation of the weak dependent components of the fractionally integrated processes. We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions that are standard in the literature and, in particular, being able to consider simultaneously (asymptotically) stationary or nonstationary processes. Relatively weak conditions on the corresponding short-run and memory parameter estimates are assumed. The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree. Finite-sample properties of the tests are analyzed by means of a Monte Carlo experiment.We thank Helmut Lütkepohl and two referees for helpful comments and suggestions. We also thank participants at the NSF/NBER Time Series Conference at the University of Heidelberg, Germany, at the Unit Root and Cointegration Testing Conference at the University of Algarve, Faro, Portugal, and seminar participants at the Universidad de Navarra and Ente Luigi Einaudi for helpful comments. Javier Hualde's research is supported by the Spanish Ministerio de Educación y Ciencia through Juan de la Cierva and Ramón y Cajal contracts and ref. SEJ2005-07657/ECON. Carlos Velasco's research is supported by the Spanish Ministerio de Educación y Ciencia, ref. SEJ2004-04583/ECON.