Palgrave Handbook of Econometrics: Applied Econometrics

J. Carlos Escanciano and Ignacio N. Lobato 991

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

Nonlinear IPRF plot

Lag j

KS (

j)

Figure 20.6 IPRF for the daily yen
Top graph is the heteroskedasticity robust autocorrelation plot. Bottom graph is the IPRF plot.

whereFis the cumulative distribution function (c.d.f.) ofYt. The measureγj,w(x)
can be viewed as a generalization of the usual autocovariance to measure the con-
ditional mean dependence in a nonlinear time series framework. It can easily be
estimated from a sample. For example, the IPAF can be estimated by:

̂γj,ind(x)=

1

n−j

∑n

t= 1 +j

(Yt−Y) 1 (Yt−j≤x). (20.7)

Moreover, as proposed by Escanciano and Velasco (2006b), nonlinear correlo-
grams can be used to formally assess the nonlinear dependence structure in the
conditional mean of the series. These authors define the KS test statistic as:

KS(j):= sup

x∈[−∞,∞]d

∣∣

∣∣(n−j)

1

(^2) ̂γj,ind(x)
∣∣
∣∣= max
1 +j≤t≤n
∣∣
∣∣(n−j)
1
(^2) ̂γj,ind(Yt−j)
∣∣
∣∣.
The asymptotic quantiles ofKS(j)under the MDH can be approximated via a wild
bootstrap approach. With these bootstrap critical values we can calculate uniform
confidence bands for̂γj(x)and the significance ofγj(x)can be tested. The plot
of a standardization ofKS(j)against the lag parameterj≥1 can be viewed as a
generalization of the usual autocovariance plot in linear dependence to nonlinear
conditional mean dependence. Escanciano and Velasco (2006b) call this plot the
integrated pairwise regression function (IPRF) plot.