Model-free evaluation of directional predictability in foreign exchange markets, страница 23

Notes:

1.  Cc1, currency 1; Cc2, currency 2.

2.  GCS tests are asymptotically one-sided N(0,1) tests and thus upper-tailed asymptotic critical values may also be used, which are 1.65 and 2.33 at the 5% and 1% levels, respectively. M1,l represents test statistics on the martingale test, serial correlation test, ARCH-in-mean test, skewness-in-mean test and kurtosis-in-mean test for l D 0,...4, respectively. 3. The direction of joint negative changes is defined by Z_t c, where return of currency k is Y_kt D 100lnS_kt/S_kt1 for k D 1,2.

4. A preliminary bandwidth p is crucial to run GCS tests. We have computed GCS test statistics for p D 11,...,50, but reported only for the value of p D 21 to save space.

two sections list the GCS test statistics using past individual returns of each currency (hereafter, ‘individual returns’). For each section, we have a total of 15 panels corresponding to 15 sample joint changes (obtained by combination of six different currencies). Subsequently, each panel contains the following GCS tests performed on three different thresholds (c D 0,0.5,1) respectively:^[22](i) the omnibus test that checks whether the direction of joint changes is predictable using past joint returns, denoted M_ZY1Y₂1,0,0, and using past individual returns, denoted M_ZYk  2; (ii) the derivative tests that examine whether directional predictability of joint changes can be explained by the level, volatility, skewness and kurtosis of joint returns, denoted M_ZY,l,l for l D 1,2,3,4, and of individual returns, denoted M_ZYk 1,l for l D 1,2,3,4 and k D 1,2; (iii) the tests that check whether the directions of past joint returns and past individual returns can be used to predict the direction of future joint changes, which are respectively denoted as

M_ZZY1Z_Y21,0,0 and M_ZZYk 1,0 for k D 1,2.

First, M_ZY1Y₂1,0,0 shows that, although there is little evidence that the direction of joint negative changes with zero threshold is predictable using past joint returns, there exists strong evidence that the direction of large joint negative changes c D 0.5,1 is predictable using past joint returns of the two currencies. Likewise, M_ZY11,0 and M_ZY show equally strong evidence that past individual returns of each currency are useful in predicting the direction of large joint negative changes c D 0.5,1. Further, the values of these omnibus test statistics M_ZY1Y₂1,0,0, M_ZY11,0 and M_ZY are generally monotonically increasing in threshold level c, implying that the direction of joint negative changes in the spot market becomes more easily predictable as we consider larger downside co-movements.

It is interesting to note that we have documented significant results whenever joint negative changes are formed by two underlying individual currency returns for which their directional predictability for single negative changes are quite strong.^[23] Moreover, they can be predicted (and explained) equally by past joint returns and past individual returns. On the other hand, when only one currency has predictive power for the direction of joint negative changes, the GCS tests based on joint returns of two currencies show somewhat mixed results. Thus, we may argue that directional dependence induced by individual currency returns plays an important role in explaining directional predictability of joint changes.^[24]

Turning to the remaining GCS tests in Table IV, we find that the sources of directional predictability of joint changes differ from what we have observed for those of single currency changes. More specifically, we can see from M_ZY1Y₂1,l,l for l D 1,2,3,4 and M_ZZY1Z_Y21,0,0 that: (i) in general, the levels of joint returns are most useful in predicting the direction of future joint negative changes; (ii) the volatilities and directions of joint returns are very useful in predicting the direction of large joint negative changes (c D 0.5,1); (iii) in the spot market, the skewness and kurtosis of joint returns are useful in predicting the direction of large joint negative changes (c D 0.5,1).^[25] Next, M_ZYk 1,l for l D 1,2,3,4 and M_ZZYk 1,0 of currency k,k D 1,2, suggest that: (i) the volatility of past individual returns is most helpful in predicting the direction of joint negative changes (as we observed in the previous single currency study); (ii) the level, skewness, kurtosis and direction of past individual returns are also useful sometimes, but are insignificant in many cases.^[26]