Palgrave Handbook of Econometrics: Applied Econometrics

330 Economic Cycles

A multivariate version of the GMM test for severalS-series is presented by Harding
and Pagan (2006), but with the GMM estimator for the bivariate case based only
on the moment conditions:

E[Sjt]=μj,j=1, 2 (7.23)

E

[

(S 1 t−μ 1 )(S 2 t−μ 2 )

√

μ 1 ( 1 −μ 1 )μ 2 ( 1 −μ 2 )

−ρs

]

=0. (7.24)

Stack the above three moment conditions into a 3x1 vector,ht(θ,S 1 t,S 2 t), such
that:

ht(θ,S 1 t,S 2 t)′=

[

S 1 t−μ 1 ,S 2 t−μ 2 ,

(S 1 t−μ 1 )(S 2 t−μ 2 )

√

μ 1 ( 1 −μ 1 )μ 2 ( 1 −μ 2 )

−ρs

]

, (7.25)

with parameter vectorθ′=[μ 1 ,μ 2 ,ρS], and take the average:

g(θ,{S 1 t,S 2 t}Tt= 1 )=

1

T

∑T

t= 1

ht(θ,S 1 t,S 2 t). (7.26)

The covariance matrix of

√

Tg(θ,{S 1 t,S 2 t}Tt= 1 )is consistently estimated by:

V= 0 +

∑m

k= 1

[

1 −

k

m+ 1

]

[k+′k], (7.27)

where:

k=

1

T

∑T

t=k+ 1

ht(θ,S 1 t,S 2 t)ht−k(θ,S 1 t,S 2 t)′. (7.28)

Harding and Pagan (2006) recommend the window width,m, to be the integer
part of(T− 1 )^1 /^3.
Letθ 0 ′=[μ 1 ,μ 2 ,0]be the restricted parameter vector forH 0 :ρS=0, with no
common cycle betweenS 1 andS 2 under this null hypothesis. The test statistic:

WSNS=

√

Tg

(

θ 0 ,{S 1 t,S 2 t}Tt= 1

)′

V−^1

√

Tg

(

θ 0 ,{S 1 t,S 2 t}Tt= 1

)

, (7.29)

follows an asymptotic chi-square distribution with one degree of freedom. Sub-
stituting sample means, ̂μ 1 and̂μ 2 , and the sample correlation, rs, for their
population counterparts does not affect the asymptotic distribution ofWSNS.
Substituting sample moments for population moments reducesWSNSto:

WSNS=T(rs− 0 )̂v−^1 (rs− 0 ), (7.30)

wherêvis the lower right-hand element of̂V. An equivalent test statistic is the

asymptotict-ratio,tSNS=T^1 /^2 ̂v−^1 /^2 rsaN(0, 1).
Closely related tests are the market-timing test of Pesaran and Timmermann
(1992) and Pearson’s chi-square test for independence. For instance, Artis,