996 Testing the Martingale Hypothesis
Table 20.5 Testing the MDH of exchange rates returns
BootstrapP-values. Generalized spectral testsDaily Weekly
Euro Pound Can Yen Euro Pound Can YenD^2 n,exp 0.023 0.450 0.680 0.913 0.670 0.123 0.360 0.586
D^2 n,ind 0.016 0.343 0.640 0.923 0.800 0.253 0.526 0.524wherêγj,indis given in (20.7). The test statisticD^2 n,expis based onw 0 (Yt−j,x)=
exp(ixYt−j)and the standard normal c.d.f. as the weighting functionW(·), where-
asD^2 n,indis based onw 0 (Yt−j,x)= 1 (Yt−j≤x)and the empirical c.d.f. as the
functionW.
We have applied these two generalized spectral distribution based tests to our
exchange rate data. The results are reported in Table 20.5 and support our previous
conclusions. Only the MDH for the daily euro exchange rate is rejected.
20.5 Related hypotheses
In this chapter we have considered testing the MDH which, in statistical terms,
just implies that the mean of an economic time series is independent of its past.
The procedures studied in this chapter can be straightforwardly applied to testing
the following generalization of the MDH:
H 0 :E[Yt|Xt−1,Xt−2,...]=μ, μ∈R,whereYtis a measurable real-valued transformation ofXtandμ=E[Yt]. This null
hypothesis, which is referred to as the generalized MDH, contains many interesting
testing problems as special cases. For instance, whenYtis a power transformation
ofXt, this null hypothesis implies constancy of conditional moments. The leading
case in financial applications is whereYt=X^2 t, because whenXtfollows an m.d.s.,
this null hypothesis means that there is no volatility in the seriesXt, i.e.,Xtis con-
ditionally homoskedastic. The casesYt=Xt^3 orYt=Xt^4 would, respectively, test
for no dynamic structure in the third moment (conditionally constant skewness)
and fourth moment (conditionally constant kurtosis) (see, e.g., Bollerslev, 1987;
Engle and González-Rivera, 1991). Another relevant case is whenYt= 1 (Xt>c),
c∈Rd. In this case, the null hypothesis represents no directional predictability
(see, e.g., Linton and Whang, 2007). Another situation of interest occurs when the
null hypothesis is the equality of the regression curves of two random variables,
X 1 tandX 2 t, say; in this case,Yt=X 1 t−X 2 t,μ=0 (see Ferreira and Stute, 2004,
for a recent reference).
Note also that most of the procedures considered in this chapter are also appli-
cable for testing the null hypothesis that a general dynamic nonlinear model is