Palgrave Handbook of Econometrics: Applied Econometrics

J. Carlos Escanciano and Ignacio N. Lobato 977

series that can present some form of dependence. Obviously, a white-noise series
is neither necessarily independent nor m.d.s., since dependence can be reflected
in other aspects of the joint distribution such as higher-order moments. The dis-
tinction between these three concepts has been stressed recently in econometrics.
In fact, during the last few years a variety of models designed to reflect nonlin-
ear dependence have been studied in the econometrics literature. For instance, in
empirical finance, ARCH and bilinear models have been widely studied (see Bera
and Higgins, 1993, 1997, and Weiss, 1986, for a comparison). These models are
suitable for reflecting the nonlinear dependence structure found in many financial
series.
Tests for white noise have been proposed both in the time domain and in the
frequency domain. The time domain has mainly, but not exclusively, focused on
a finite number of lags, while the frequency domain has been more suitable for
addressing the infinite-dimensional case.

20.3.1 Tests based on a finite-dimensional conditioning set

In the time domain the most popular test (apart from the Durbin–Watson, which
is designed to test for lack of first-order serial correlation using regression residuals)
has been the Box–Pierce (Box and Pierce, 1970) portmanteauQptest. TheQptest is
designed for testing that the firstpautocorrelations of a series (possibly residuals)
are zero. The numberpcan be considered to be fixed or to grow with the sample
sizen. In this section we will assume thatpis fixed.
Suppose that we observe raw data{Yt}nt= 1. Thenγjcan be consistently estimated
by the sample autocovariance:

̂γj=(n−j)−^1

∑n

t= 1 +j

(Yt−Y)(Yt−j−Y),

whereYis the sample mean, and we also introducêρj=̂γj/̂γ 0 to denote the
jth-order autocorrelation. TheQpstatistic is just:

Qp=n

∑p

j= 1

̂ρ^2 j,

but it is commonly implemented via the Ljung and Box (1978) modification:

LBp=n(n+ 2 )

∑p

j= 1

(n−j)−^1 ̂ρ^2 j.

Note thatQp(orLBp)only takes into account the linear dependence up to the lagp.
Whenpis considered fixed, theQptest statistic applied to independent data follows

aχ^2 pasymptotic null distribution, since the asymptotic covariance matrix of the
firstpautocorrelations of an independent series is the identity matrix. Hence, it is

useful to writeQp=

(√

n̂ρ

)′

I−^1

(√

n̂ρ

)

, wherêρ=(̂ρ 1 , ...,̂ρp)′.