An easy test for two stationary long processes being uncorrelated via AR approximations

This paper proposes an easy test for two stationary autoregressive fractionally integrated moving average (ARFIMA) processes being uncorrelated via AR approximations. We prove that an ARFIMA process can be approximated well by an autoregressive (AR) model and establish the theoretical foundation of Haugh's (1976) statistics to test two ARFIMA processes being uncorrelated. Using AIC or Mallow's Cp criterion as a guide, we demonstrate through Monte Carlo studies that a lower order AR(k) model is sufficient to prewhiten an ARFIMA process and the Haugh test statistics perform very well in finite sample. We illustrate the methodology by investigating the independence between the volatility of two daily nominal dollar exchange rates-Euro and Japanese Yen and find that there exists "strongly simultaneous correlation" between the volatilities of Euro and Yen within 25 days.