Palgrave Handbook of Econometrics: Applied Econometrics

J. Carlos Escanciano and Ignacio N. Lobato 995

r(

j)

KS (

j)

–0.2

–0.1

Lag j

Lag j

0 5 10 15 20 25 30 35

0

0.1

0.2

Autocorrelogram

0 5 10 15 20 25 30 35

0

0.5

1

1.5

Nonlinear IPRF plot

Figure 20.10 IPRF for the weekly yen
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

In order to evaluate the distance ofSn(λ,x)from zero, a norm has to be chosen.
One norm considered in practice is the Cramér–von Mises norm:

D^2 n,w=

∫

R

∫^1

0

∣∣

∣Sn,w(λ,x)

∣∣

∣

2

W(dx)dλ=

n∑− 1

j= 1

(n−j)

1

(jπ)^2

∫

R

∣∣

∣̂γj,w(x)

∣∣

∣

2

W(dx), (20.12)

whereW(·)is a weighting function satisfying some mild conditions.D^2 n,whas the
attractive convenience of being free of choosing any smoothing parameter or ker-
nel, and it has been documented to deliver tests with good power properties (cf.
Escanciano and Velasco, 2006a, 2006b).
Among the members of this class of test statistics, the most common choices are:

D^2 n,exp=̂σ−^2

n∑− 1

j= 1

(n−j)

1

(jπ)^2

∑n

t=j+ 1

∑n

s=j+ 1

(Yt−Yn−j)(Ys−Yn−j)exp(−0.5(Yt−j−Ys−j)^2 ),

and:

D^2 n,ind=̂σ−^2

n∑− 1

j= 1

(n−j)

n(jπ)^2

∑n

t= 1

̂γj^2 ,ind(Xt),