Model-free evaluation of directional predictability in foreign exchange markets, страница 11

¹⁶  As different choices of the derivative orders m,l yield tests of various hypotheses, different WÐ may be selected depending on which hypothesis is of interest. For the omnibus test M_ZY1,0, we put W, where W is the N(0, 1) CDF. For the separate tests 1, we put υÐ, where υÐ is the Dirac delta function; namely, υu D 0 for all u 6D 0 andυudu D 1. For further discussion, see Hong (1999).

¹⁷  Alternatively, one could test directional predictability by testing the i.i.d. property for. Because the direction indicator Z_tc is a Bernoulli random variable taking value 0 or 1, it is independent of I_ttc is not predictable using I_t1. Thus, if evidence against i.i.d. is found for, one can conclude that the direction of returns is predictable using the past history of the return directionsc,Z_t2c,...g. See Hong and Chung (2006) for further discussion.

Finally, it is straightforward to test whether interest rate differentials fID_tg are useful in predicting the direction of foreign exchange rate returns fZ_tg. We will repeat the above evaluation measures, but with change of argument Y_t D ID_t. Accordingly, we denote M_ZID1,0 as an omnibus test for ₀ of (4), and M_ZID1,l with l D 1,2,3,4 and M_ZZID1,0 as separate tests to check whether fZ_tg is predictable using the level, volatility, skewness, kurtosis and directions of past interest rate differentials, respectively.

3.2.  Tests for the Direction of Joint Changes in Two Currencies

Our aim is now to gauge directional predictability of joint changes in two currencies. Intuitively, previous measures f_ZYω,u,v and f_ZZω,u,v cannot be directly applicable when there are more than two variables involved (as is the case when we explore the directional predictability of joint changes in two currencies using their return series fY_1t,Y_2tg).^[14] For this purpose, we will use the multivariate generalized cross-spectral density below.

Suppose we have a strictly stationary time series process fZ_t,Y_1t,Y_2tg, and define the generalized cross-covariance function between fZ_tg and_j,Y_j,j > 0g as

                                                      _ZY1Y₂,ju,cove^iuZt,eⁱ,                                       j D 0,š1,...                            17

where u, 1. By the Fourier transform of _ZY1Y₂,ju,v, , we readily obtain the generalized cross-spectral density between fZ_tg and_j,Y_j,j > 0g:

                                                      f_ZY1Y₂ω,u,_ZY1Y₂,ju,v, e^ijω,                                ω             ,]                          18

2 jD1

Like _ZYu,v and _ZZu,v, because _ZY 0 for all u, if and only if fZ_tg and_j,Yare mutually independent, _ZY,  can capture any type of pairwise cross-dependence between fZ_tg and_j,Y, and so is f_ZY1Y₂ω,u,v, . As a result, we can use f_ZY1Y₂ω,u,v,  to explore how Z_t depends on the entire past history of two currency returns_j,j > 0g.

When EjZ_tj^2m < 1 and EjY_1tj^2l C jY_2tj^2l < 1, we can introduce the generalized crossspectral density derivative between fZ_tg and_j,Y_j,j > 0g by defining

                      0,m,l,l^∂mC2l                                                                                          1            m,l,l                            ijω

fZY1Y2 ω,u,l fZY1Y2ω,u,                                                                 2                   ZY1Y2,ju,v, e             ,         m,l ½ 0

jD1

19 As before, our test statistics for the direction of joint changes will be based on comparison via the quadratic form between two cross-spectral derivative estimators f^O _ZY^0,1₁^,l,l_Y2 ω,u,v,  and f^O _ZY^0,1,l,l_{Y ,0}ω,u,v, , where the latter is implied by the null hypothesis ₀ of no directional

1 2

predictability:

QO 1,l,l D T^,l,l_Y2 ω,ZY0,1₁,l,lY₂,0ω,dW

T1

                                j/pT                                                                    dW                                      20

jD1

Accordingly, we have the test statistic