Model-free evaluation of directional predictability in foreign exchange markets, страница 14

4.  denote the domestic (US) and foreign risk-free interest rates, respectively.

for c D 0,0.5,1, in units of the sample standard deviation of fY_tg.^[18] Here, two types of indicator function are designed to examine dynamic characteristics of directional movement in up and down markets,^[19] while three threshold values are used to capture different magnitudes of changes in returns. In this paper, we mainly focus on pairwise cross-dependencies between Z_tc and two key variables: past returns and past interest rate differentials, where r_t is the domestic (US) risk-free interest rate and r_t^Ł is the foreign risk-free interest rate at time t. We further rescale interest rate differentials centered at 0 to synchronize the levels of interest rate differentials across different countries.

We first examine directional predictability using the past history of fY_tg. Table II reports the test statistics M_ZY1,l for l D 0,1,2,3,4 and M_ZZ1,0 with the preliminary lag order p D 21 and the Bartlett kernel.^[20] Note, for comparison, that GCS tests are asymptotically one-sided N(0, 1) tests and thus upper-tailed N(0, 1) critical values should be used, which are 1.65 and 2.33 at the 5% and 1% significance levels, respectively.

The first top panel reports the omnibus M_ZY1,0 statistic, checking whether the directions of each currency return is predictable using its own past returns _j,j > 0g. For all individual currency returns in both spot and futures markets, there exists strong evidence of directional predictability. Except for FSF and FDM, the M_ZY1,0 statistic value becomes larger as threshold c increases, suggesting that the directions of large returns are easier to predict than the directions of small returns using past returns. Comparing the spot and futures markets, we find that the directions of the returns in the futures market are generally easier to predict with zero threshold c D 0. In contrast, when c D 1, the evidence is stronger for those in the spot market in most cases. Further, there is no clear evidence that the direction of negative returns is easier to predict than that of positive returns, using past returns. Among other things, the directions of AD, CD and BP in both spot and futures markets (respectively, FAD, FCD and FBP) are considerably easier to predict, especially with large thresholds (c D 0.5,1).

The remaining panels in Table II examine possible sources of the documented directional predictability. We report the test statistics M_ZY1,l for l D 1,2,3,4 and M_ZZ1,0. These test statistics can tell us to what extent past returns contain useful information for predicting the direction of future returns. Specifically, each test statistic shows whether the direction of negative and positive returns (left to right) can be predicted using the level, volatility, skewness, kurtosis and the direction of past returns (top to bottom), respectively. As shown in M_ZY1,1, the level of past returns Y has not been shown to be very useful in predicting the directions of individual currency returns, because no clear pattern of statistical significance emerges for the M_ZY1,1 test. In contrast, M_ZY1,2 shows strong evidence that past volatility is a valuable source of information about directional predictability of individual currency returns, particularly with large thresholds c D 0.5,1. Like M_ZY1,0, the test statistic M_ZY1,2 is monotonically increasing in threshold level c, except for the direction of positive changes in FSF. Moreover, using past volatility, it is generally easier to predict the direction of negative returns than that of positive returns, and the directions of the returns in the spot market than those in the futures market.