Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 683

(iii) Group mean tests (nonparametric corrections)

group meanρ-test:N−^1 /^2

∑N

i= 1

(

T−^1

∑T

t= 2

uˆi,t− 1 uˆi,t−ψi

)

(

T−^2

∑T

t= 2

uˆ^2 i,t− 1

)

group mean t-test:N−^1 /^2

∑N

i= 1

(

T−^1

∑T

t= 2

uˆi,t− 1 uˆi,t−ψi

)

ωˆυ,i

(

T−^2

∑T

t= 2

uˆ^2 i,t− 1

) 1 / 2

(iv) Group mean tests (parametric corrections)

group mean t-test–parametrically corrected:N−^1 /^2

∑N

i= 1

(

T−^1

∑T

t=Ki+ 2

ζˆ1,i,tζˆ2,i,t

)

σˆθ,i

(

T−^2

∑T

t=Ki+ 2

ζˆ^2

2,i,t

) 1 / 2.

The mean and variance correction terms (for largeNandT)for each of these seven
tests depend upon the deterministic specification considered, as well as upon the
dimension of thexi,tvector. Once recentered and scaled by the appropriate mean
and variance corrections, the standardized statistics tend in the limit to theN(0, 1)
density (under sequential convergence,T→∞,N→∞). Hlouskova and Wagner
(2008) contains a large set of finite sample and asymptotic correction factors for
up to 12 regressors.

13.3.1.2 Some general remarks

It is worth re-emphasizing that the testing principles derived earlier for unit root
tests in panels carry over in direct ways to the testing for cointegration, with the
obvious (but by no means trivial) embellishments of endogeneity correction, and
the need for methods that estimate the cointegrating vector efficiently and consis-
tently when required. A class of tests due to Kao (1999) is constructed under the
same general framework as described above for the Pedroni tests, while Wester-
lund (2005) develops two simple nonparametric tests, one against a homogeneous
alternative whilst the other is a group mean test against a heterogeneous stationary
alternative.
Wagner and Hlouskova (2007), in a companion study to Hlouskova and Wagner
(2006), report the results of an extensive simulation exercise, one part of which
is devoted to looking at the behavior of the Pedroni and Westerlund tests. The
study explores the size and power properties of the tests under the baseline case
where cross-sectional independence is imposed in the DGP, but also reports results
when cross-sectional independence is violated both via a correlation structure, as
described in section 13.2.2.1 (or a slight modification, where the correlation matrix