An Easy Test for Independence between Two Stationary Long Memory Processes via AR Approximations, With An Application to the Volatility of Foreign Exchange Rates

This paper proposes an easy test for independence between two stationary autoregressive fractionally integrated moving average (ARFIMA) processes via AR approximations. We prove that an ARFIMA (p, d, q) process, φ(L)(1 − L)yt = θ(L)et, d ∈ (0, 0.5), where et is a white noise, can be approximated well by an autoregressive (AR) model and establish the theoretical foundation of Haugh’s (1976) statistics to test the independence between two ARFIMA processes. Using Haugh statistic can also avoid the issues of spurious regression induced by the long memory processes considered by Tsay and Chung (2000). We also provide the guidance of determination of the order of AR(k) model fitted to long memory processes. Monte Carlo experiments are conducted to investigate the finite sample properties of our analytical results. The simulations confirm the desirability of using the Haugh test statistics after AR pre-whitening of ARFIMA processes to test for independence in finite sample. We illustrate the use of our methodology by investigating the independence between the volatility of two daily nominal dollar exchange rates-Euro and Japanese Yen. We find that there exists ” strongly simultaneous correlation ” between the volatilities of Euro and Yen within 30 days.