Palgrave Handbook of Econometrics: Applied Econometrics

J. Carlos Escanciano and Ignacio N. Lobato 989

0 5 10 15 20 25 30 35

–0.1

–0.05

0

0.05

0.1

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 20.3 IPRF for the daily euro
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

Using a different methodology based on the generalized spectral density
approach of Hong (1999), Hong and Lee (2003) proposed an MDH bootstrap test
(see also Hong and Lee, 2005). Tests based on the generalized spectral density
involve three choices: a kernel, a bandwidth parameter, and an integrating mea-
sure; and, in general, statistical inferences are sensitive to these choices. This fact
motivated Escanciano and Velasco (2006a, 2006b) to propose testing the MDH by
means of a generalized spectral distribution function.
The generalized spectral approach is based on the fact that the MDH implies that:

H 0 :γj,w(x)= 0 ∀j≥1, for allx, (20.6)

whereγj,w(x)=E[(Yt−μ)w 0 (Yt−j,x)]and wherew 0 (Yt−j,x)is any of the para-
metric functions of the previous section. The generalized spectral approach of
Hong is based on the choicew 0 (Yt−j,x)=exp(ixYt−j). Escanciano and Velasco
(2006a) considered the latter choice, while Escanciano and Velasco (2006b) used
w 0 (Yt−j,x)= 1 (Yt−j≤x), and called the measureγj,ind(x)=E[(Yt−μ) 1 (Yt−j≤x)]
the integrated pairwise autoregression function (IPAF). The name follows from the
fact that:

γj,ind(x)=E[(Yt−μ) 1 (Yt−j≤x)]=

∫x

−∞

E[Y−μ|Yt−j=z]F(dz),