Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 687

for theith unit the fraction which minimizes the sequence of the individual
t-statistics. That is, for uniti,

Tˆb,i= arg min

λi∈[0.15,0.85]

tρˆi(λi).

The limiting distribution of infλi∈[0.15,0.85]tρˆi(λi)is shown by Gregory and Hansen
(1996) not to depend on the break fraction parameters and it is for this reason that
the minimization is undertaken over the sequence of break fractions in uniti. The
procedure is repeated for all unitsi, leading to a sequence of “estimated” break
fractions for theNunits, denoted in vector notation asλˆ=(λˆ 1 ,λˆ 2 ,...,ˆλN)′.
Finally, the pooled test (in the spirit of Pedroni, 1999, 2004), which allows for
breaks under the alternative hypothesis, is given by:

N−^1 /^2 Zˆt

N,T

(λ)ˆ =N−^1 /^2

∑N

i= 1

tρˆi(λˆi).

The asymptotic distribution ofN−^1 /^2 ZˆtN,T(λ)ˆ is given by the theorem below. It

is important to note that in this framework (apart from the restriction to cross-
sectional independence, which is lifted in the following sub-section) a high degree
of heterogeneity is allowed across the units, since the cointegrating vector, the
short-run dynamics and the break date are all allowed to differ among units. In
the spirit of much of this literature, and in particular of Pedroni (1999, 2004),
the panel test statistics are shown to converge to standard normal distributions
once they have been properly standardized. The correction terms, of course, differ
from those tabulated by Pedroni (2004) since the models are considerably more
complicated due to the presence of breaks at unknown points of time across the
units, but the principles involved remain the same. The following result may now
be shown:

Theorem (Banerjee and Carrion-i-Silvestre, 2007, Theorem 1): Letand!denote the
mean and variance for the vector Brownian motion functional:

Υ′=

(

inf

λi∈%

∫ 1

0

Q(λi,s)dQi(λi,s)

[∫

1

0

Q(λi,s)^2 ds

]− 1

, inf

λi∈%

∫ 1

0

Q(λi,s)dQi(λi,s)×

[∫

1

0

Q(λi,s)^2 ds·( 1 +ρ(λi)′D(λi)ρ(λi))

]− 1 / 2 )

.

Q(λi,s)andρ(λi)are functions of vector Brownian motions and the deterministic compo-
nents and D(λi)depends on the model chosen (see Gregory and Hansen, 1996, for details).
Then, as T→∞,N→∞in sequence, under the null hypothesis of no cointegration the

asymptotic distribution of the statistic ZˆtN,T(λ)ˆ is given by N−^1 /^2 ZˆtN,T(λ)ˆ − 2

√

N⇒

N(0,! 2 ).

Several remarks are appropriate in connection with this theorem. First, we may
derive similar theorems for the remaining statistics proposed by Pedroni (1999,