Palgrave Handbook of Econometrics: Applied Econometrics

978 Testing the Martingale Hypothesis

Note, however, that when the series present some kind of nonlinear dependence,
such as conditional heteroskedasticity, this asymptotic null covariance matrix is no
longer the identity. In fact, denotingρ=(ρ 1 , ...,ρp)′, for a general time series the
asymptotic distribution of

√
n(̂ρ−ρ)isN(0,T), where thep×pmatrixThas as
(i,j)th element (see, e.g., Romano and Thombs, 1996),

γ 0 −^2 (cij−ρic 0 j−ρjc 0 i+ρiρjc 00 ),

where, fori,j=0, 1, ...,p,

cij=

∑∞

d=−∞

{

E

[

(Yt−μ)(Yt−i−μ)(Yt+d−μ)(Yt+d−j−μ)

]

−E

[

(Yt−μ)(Yt−i−μ)

]

E

[

(Yt+d−μ)(Yt+d−j−μ)

]}

.

Under alternative assumptions the matrixTcan be simplified and this will lead
to several modified versions of the Box–Pierce statistic. When this matrix is still
diagonal, as happens under m.d.s. and additional moment restrictions, which, for
instance, are satisfied by Gaussian GARCH models and many stochastic volatil-
ity models, the natural approach is to robustify theQpby standardizing it by a
consistent estimate of its asymptotic variance, i.e.,

Qp∗=n

∑p

j= 1

̂ρ^2 j

τj

,

where:

τj=

1

̂γ 02

∑n

t= 1 +j

(Yt−Y)^2 (Yt−j−Y)^2.

We have followed Lobato, Nankervis and Savin’s (2001) notation and denoted the
robustifiedQpbyQp∗. This statistic has appeared in different versions (see, e.g.,
Diebold, 1986; Lo and MacKinlay, 1989; Robinson, 1991; Cumby and Huizinga,
1992; Bollerslev and Wooldridge, 1992; Bera and Higgins, 1993). TheQp∗statistic
(or its Ljung–Box analog) should be routinely computed for financial data instead
of the standardQp(or theLBp). However, this is not typically the case (see Lobato,
Nankervis and Savin, 2001, for details).
For the general case, the asymptotic covariance matrix of the firstpautocorre-
lations is not a diagonal matrix. Hence, for this general case both theQpand the
Qp∗tests are invalid. However, under m.d.s. the matrixTcan be greatly simplified

so that itsij-th element takes the formE[(Yt−μ)^2 (Yt−i−μ)(Yt−j−μ)], which
can easily be estimated using its sample analog. This is the approach followed by
Guo and Phillips (2001). For the general case, which includes m.d.s. and non-
m.d.s. processes, the asymptotic covariance matrix of the firstpautocorrelations is
a complicated non-diagonal matrix. Hence, for this general case, the literature has
proposed the following two modifications of theQptest. The first is to modify the