Exact local Whittle estimation of fractionally cointegrated systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. We propose a two-step procedure that accommodates both (asymptotically) stationary (δ<1/2)$(\delta <1/2)$ and nonstationary (δ≥1/2)$(\delta \ge 1/2)$ stochastic trend and/or equilibrium error. A tapered version of the local Whittle estimator of Robinson (2008) is used as the first-stage estimator, and the second-stage estimator employs the exact local Whittle approach of Shimotsu and Phillips (2005). The consistency and asymptotic distribution of the two-step estimator are derived. The estimator of the memory parameters has the same Gaussian asymptotic distribution in both the stationary and the nonstationary case. The convergence rate and the asymptotic distribution of the estimator of the cointegrating vector are affected by the difference between the memory parameters. Further, the estimator has a Gaussian asymptotic distribution when the difference between the memory parameters is less than 1/2.