Palgrave Handbook of Econometrics: Applied Econometrics

992 Testing the Martingale Hypothesis

0 5 10 15 20 25 30 35

–0.2

–0.1

0

0.1

0.2

Autocorrelogram

Lag j

r(

j)

0 5 10 15 20 25 30 35

0

0.5

1

1.5

2

Nonlinear IPRF plot

Lag j

KS (

j)

Figure 20.7 IPRF for the weekly euro
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

In Figures 20.3–20.10 we plot the IPRF for our exchange rate returns. The com-
mon feature of these graphs is the lack of dependence in the exchange rates, both
linear and nonlinear. Only a few isolated statistics seem to be significant, but the
evidence is very weak. It seems that the IPRF supports the MDH for these datasets.
We now describe a generalized spectral approach to enable us to consider simulta-
neously all the nonlinear measures of dependence{γj,w(·)}. Defineγ−j,w(·)=γj,w(·)
forj≥1, and consider the Fourier transform of the functionsγj,w(x),

fw(&,x)=

1

2 π

∑∞

j=−∞

γj,w(x)e−ij& ∀&∈[−π,π]. (20.8)

Note thatfw(&,x)is able to capture pairwise non-martingale difference alter-
natives with zero autocorrelations. Under the MDH, the conditionf0,w(&,x)=

( 2 π)−^1 γ 0 (x)holds, i.e., the generalized spectral densityfw(&,x)is flat in&. Hong
(1999) proposed the estimators:

̂fw(&,x)=^1

2 π

n∑− 1

j=−n+ 1

(

1 −

∣∣

j

∣∣

n

) 1

2

k

(

j

p

)

̂γj,exp(x)e−ij&,