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 MZY1,0, 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 and
υudu 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 Ztc
is a Bernoulli random
variable taking value 0 or 1, it is independent of It
tc is not predictable using It1. 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 directions
c,Zt2c,...g. See Hong and Chung (2006) for
further discussion.
Finally, it is straightforward to test whether interest
rate differentials fIDtg
are useful in predicting the direction of foreign exchange rate returns fZtg. We will repeat
the above evaluation measures, but with change of argument Yt D IDt. Accordingly, we
denote MZID1,0 as an omnibus test for 0 of (4), and MZID1,l with l D 1,2,3,4 and MZZID1,0 as separate tests to check whether fZtg is predictable
using the level, volatility, skewness, kurtosis and directions of past interest
rate differentials, respectively.
Our aim is now to gauge directional predictability of joint changes in two currencies. Intuitively, previous measures fZYω,u,v and fZZω,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 fY1t,Y2tg).[14] For this purpose, we will use the multivariate generalized cross-spectral density below.
Suppose we have a strictly stationary time series process fZt,Y1t,Y2tg,
and define the generalized cross-covariance function between fZtg andj,Yj,j > 0g as
ZY1Y2,ju,coveiuZt,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 fZtg andj,Yj,j > 0g:
fZY1Y2ω,u,ZY1Y2,ju,v,
eijω, ω ,] 18
2 jD1
Like ZYu,v and
ZZu,v, because ZY
0 for all u, if
and only if fZtg
andj,Yare mutually independent, ZY, can capture any type of pairwise
cross-dependence between fZtg
andj,Y, and so is fZY1Y2ω,u,v, . As a result, we can use fZY1Y2ω,u,v, to explore how Zt depends on the
entire past history of two currency returnsj,j > 0g.
When EjZtj2m < 1 and EjY1tj2l C jY2tj2l < 1, we can introduce the generalized
crossspectral density derivative between fZtg
andj,Yj,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 fO ZY0,11,l,lY2 ω,u,v, and fO ZY0,1,l,lY ,0ω,u,v, ,
where the latter is implied by the null hypothesis 0 of no
directional
1 2
predictability:
![]() |
T1
j/pT dW 20
jD1
Accordingly, we have the test statistic
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