• Fractional integration;
  • local Whittle;
  • long memory;
  • multi-variate semi-parametric estimation;
  • exchange rates
  • C14;
  • C32

This article derives a semi-parametric estimator of multi-variate fractionally integrated processes covering both stationary and non-stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi-variate local Whittle estimator of Shimotsu (2007) to cover non-stationary values of d. Consistency and asymptotic normality is shown for d ∈ (−1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.