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Model-free evaluation of directional predictability in foreign exchange markets, страница 9

are independent. Thus it can capture any type of pairwise cross-dependence between fZ_tg and _j,j > 0g over various lags (including those with zero autocorrelation). In this spirit, f_ZYω,u,v can capture various linear and nonlinear cross-dependencies.^{^[11]}Another important advantage of using the characteristic function is that it requires no moment condition on fZ_tg and fY_tg, and so it does not suffer from the potential problem when the moment condition fails, which is often found in high-frequency economic and financial time series (e.g., Pagan and Schwert, 1990). Moreover, the generalized spectrum shares a nice feature of the conventional spectral approach—it incorporates information on serial dependence from virtually all lags. This will ensure to capture serial dependence at higher lag orders, and hence enhance good power for tests against the alternatives involving a persistent dependence structure (i.e., serial dependence decays to zero slowly as j ! 1).

3.1. Generalized Cross-Spectral Derivative Tests

It is important to point out that the generalized cross-spectrum f_ZYω,u,v itself is not suitable for testing the null hypotheses ₀in (2) and (4), because the generalized spectrum f_ZYω,u,v encompasses all pairwise cross-dependencies in various conditional moments of both fZ_tg and j,j > 0g. Fortunately, f_ZYω,u,v can be differentiated to reveal possible specific patterns of cross-dependence in various conditional moments, thanks to the use of the characteristic function.

In particular, one can use the generalized cross-spectral density derivative

f_ZY0,m,lω,u, ∂u∂m_mC∂^vl _lfZYω,u, 2 jD1 _ZY,jm,lu,ijω, m,l ½ 0 7

By varying the combination of the derivative orders m,l, the generalized cross-spectral derivative f_ZY⁰^,m,lω,u,v can capture various specific aspects of cross-dependence between fZ_tg and fY_tj,j > 0g. For example, to test the null hypothesis ₀of (2): EZ_t EZ_t a.s. (with Z_tD Z_tc), we can use the (1, 0)th order generalized cross-spectral derivative

fZY0,1,0ω, ZY,jeⁱ^jω, ω ,] 8

2 jD1

where

ZY,j1,0ZY,ju, coviZ_t,eⁱ^v^Y

The measure _ZY,j¹^,00,v checks correlations between Z_tand all moments of Y_t, and is thus suitable for testing whether EZ_tEZ_t for all j.^{^[12]}

As in Hong and Chung (2006), we consider a stepwise procedure for hypothesis testing, which begins by examining directional predictability using f_ZY⁰^,1^,0ω,0,v, then proceeds for separate inferences on various sources such as time-varying conditional mean, volatility clustering and conditional skewness or other higher-order conditional moments. Once directional predictability is detected, this stepwise testing procedure will reveal useful information in making inferences on the nature of directional predictability and thus the modeling of directional forecasts.¹⁴In particular, we use the following higher-order generalized cross-spectral derivative with the choice of l D 1,2,3,4 respectively:

fZY0,1,lω, ZY,jeijω, ω ,] 9

where

ZY,j1,l l ZY,ju,cov[iZt,iY l , l ½ 1

∂u∂v

As expected, _ZY¹^,l0,0 will be proportional to cross-covariance covZ_t,Y^l and, as a consequence, f_ZY⁰^,1^,lω,0,0 for l D 1,2,3,4 can be used to test whether Z_tis predictable using the level of past changes , past volatility, past skewness and past kurtosis