Palgrave Handbook of Econometrics: Applied Econometrics

J. Carlos Escanciano and Ignacio N. Lobato 993

0 5 10 15 20 25 30 35

–0.2

0

0.1

0.2

Autocorrelogram

Lag j

0 5 10 15 20 25 30 35

0

0.5

1

1.5

2

Nonlinear IPRF plot

Lag j

r(

j)

KS (

j)

–0.1

Figure 20.8 IPRF for the weekly pound
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

and:
̂f0,w(&,x)=^1
2 π

̂γ0,w(x),

to test the MDH, wherek(·)is a symmetric kernel andpa bandwidth parameter. He
considered a standardization of anL 2 -distance using a weighting functionW(·):

L^2 2,n(p)=

π

2

∫

R

∫π

−π

n

∣∣

∣̂fw(&,x)−̂f0,w(&,x)

∣∣

∣

2

W(dx)d& (20.9)

=

n∑− 1

j= 1

(n−j)k^2

(

j

p

)∫

R

∣∣

∣̂γj,w(x)

∣∣

∣

2

W(dx).

Under the null of MDH and some additional assumptions, Hong and Lee (2005)
showed that a convenient standardization ofL^2 2,n(p)converges to a standard nor-
mal random variable. The centering and scaling factors in this standardization
depend on the higher dependence structure of the series.
Alternatively, the generalized spectral distribution function is:

Hw(λ,x)= 2

λπ∫

0

fw(&,x)d&λ∈[0, 1],