Palgrave Handbook of Econometrics: Applied Econometrics

976 Testing the Martingale Hypothesis

14-Jan-2000 26-Jul-2002 04-Feb-2005 17-Aug-2007

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Euro

14-Jan-2000 26-Jul-2002 04-Feb-2005 17-Aug-2007

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Pound

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Can

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Yen

Figure 20.2 Weekly returns of the euro, Canadian dollar (Can), the pound sterling (Pound)
and the Japanese yen (Yen) against the US dollar
Data from January 14, 2000, to August 17, 2007.

Table 20.1 Summary statistics of exchange rates returns

Daily Weekly

Euro Pound Can Yen Euro Pound Can Yen

n 908 908 908 908 382 382 382 382

Mean 0.0076 0.0113 −0.0213 0.0068 0.0738 0.0552 −0.0832 0.0352

Median 0.0000 0.0221 −0.0080 0.0279 0.0781 0.0763 −0.0864 0.0141

SD 0.5423 0.5332 0.5036 0.5670 1.3539 1.1407 0.9410 1.2525

Skewness −0.1263−0.0976 −0.0196 −0.3763 0.0540 0.0545 0.0846 −0.2945

Kurtosis 3.7602 3.4927 3.1345 5.0746 3.0555 2.9649 2.8875 3.0895

Maximum 1.9358 2.0930 1.5129 2.4519 4.4680 3.4830 2.8128 3.1835

Minimum −2.0355−2.1707 −1.7491 −2.7859−3.1636−3.2307 −2.7067 −4.3058

for somej≥1. Hence, a necessary (but not sufficient, in general) condition for the
the MDH to hold is that the time series is uncorrelated, i.e.,

γj=Cov(Yt,Yt−j)=E[(Yt−μ)Yt−j]= 0 for allj≥1, (20.3)

whereγjdenotes the autocovariance of orderj. In principle, one should test that
all autocovariances or autocorrelations are zero. However, the most employed tests
just consider that a finite number of autocorrelations are zero. As commented in
the introduction, we will address these two cases separately.
Notice that the early literature, which includes some distinguished references
such as Yule (1926), Bartlett (1955), Grenander and Rosenblatt (1957) or Durbin
and Watson (1950), essentially assumed Gaussianity and, hence, identified three
concepts: lack of serial correlation, m.d.s., and independence. In the time series
literature the term “white noise” is commonly used to denote an uncorrelated