Palgrave Handbook of Econometrics: Applied Econometrics

660 Panel Methods to Test for Unit Roots and Cointegration

Y− 1 =(y1,− 1 ,y2,− 1 ,...,yN,− 1 )

uˆi=(uˆi,1,...,uˆi,T)′

uˆ=(uˆ 1 ,uˆ 2 ,...,uˆN)

ˆe=uˆQˆ%ˆ

ωe^2 ,i=

⎛

⎝

∑∞

j= 0

di,j

⎞

⎠

2

λe,i=

∑∞

l= 1

∑∞

j= 0

di,jdi,j+l

ω^2 e=

1

N

∑N

i= 1

ω^2 e,i

φ^4 e=

1

N

∑N

i= 1

ω^4 e,i

λNe =

1

N

∑N

i= 1

λe,i.

Consistent estimators ofω^2 e,φ^4 eandλNe are provided by constructing kernel esti-
mators based on sample covariances provided by^1 T
∑
tˆei,tˆei,t+j, whereeˆi,tis the
(i,t)th element ofˆedefined above and 1≤t,t+j≤T. These are denoted byωˆe^2 ,

φˆe^4 andλˆNe, respectively.
Two test statistics, denotedMPaandMPb, follow, defined as:

MPa=

√

NT(φ∗Pooled− 1 )

√

2 φˆ^4 e

ωˆ^4 e

MPb=

√

NT(φ∗Pooled− 1 )

φˆ^2 e

√

1

NT^2

tr(Y− 1 Qˆ%ˆY−′ 1 )ωˆe,

whereφPooled∗ =

tr(Y− 1 Qˆ%ˆY′)−NTˆλNe

tr(Y− 1 Qˆ%ˆY−′ 1 )

.

Under the null hypothesisH 0 :φi= 1 ∀i, Moon and Perron (2004) show that:

MPa,MPb⇒N(0, 1),asN,T→∞withN/T→0.^19

No inference concerning the unit root behavior is conducted on the estimated
factors, since these are taken to be stationarya priori. However, under the null
hypothesis it is the accumulated factors that enter the datay.