Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 665

As in section 13.2.2.2, let:

yi=(yi,2,...,yi,T)′

ei∗=(ei∗,2,...,e∗i,T)′

F=(F 2 ,...,FT)′ ((T− 1 )×rmatrix of differenced factors

for all the units)

πi=(πi,1,...,πi,r)′ (r×1 vector of loadings of factors forith unit).

Then we may write the model (in vector notation) as:

y ̃i=fπi+zi,

wherey ̃i=yi,f=Fandzi=e∗i.

The estimated factorsfˆand their loadingsπˆiare calculated as in section 13.2.2.2,
and:

ˆzi= ̃yi−fˆπˆi; and, finally,ˆei,t=

∑t

s= 2

zˆi,s.

Since the MSB statistic is not affected by the impulse dummies, and hence by the
break fractionsγi, Bai and Carrion-i-Silvestre prove that under the null hypothesis

of a unit root and the assumption that

Tai,j

T =γiremain constant asT→∞:

MSB(i)⇒

∫ 1

0

Wi^2 (r)dr,

whereWi(r)is a standard Brownian motion independent acrossi.
The pooled statistic has the form:

Z=

√

N

MSB−0.5

1

/√

3

,

and has a limiting normal distribution, asN→∞, whereMSB=N^1

∑N
i= 1 MSBi.
The mean and variance correction terms are those appropriate for the individual
MSB(i)statistics.
A pooled Fisher-type test can also be constructed. Denoting bypithep-value of
theMSB(i)test for theith unit, then:

BCχ 2 =− 2

∑N

i= 1

logpi∼χ 22 N.

This is a result applicable to cases where the cross-section dimensionNis finite.
WhenNis large, we may also use the asymptotic approximation to the chi-squared
statistic proposed by Choi (2001), that is,

BCN=

− 2

∑N

i= 1

logpi− 2 N

√

4 N

⇒N(0, 1).