Palgrave Handbook of Econometrics: Applied Econometrics

646 Panel Methods to Test for Unit Roots and Cointegration

where individual specific serial correlation structures are allowed in (13.10), with
the assumption that theyi,tfollowAR(pi+ 1 )processes.
We start by establishing some notation, assuming for simplicity that all required
lagged observations are available. From (13.10), let:

φ ̃i=(φi,1,...,φi,pi)′

yi,− 1 =(yi,0,yi,1,...,yi,T− 1 )′

yi,−s=(yi,1−s,yi,2−s,...,yi,T−s)′,s=0, 1,...,pi

yi=yi,− 0

d 2 T=(1, 1,...,1)′

tVEC=(1, 2,...,T)′

d 3 T=(d′ 2 T,tVEC′ )

Qi,m=(dmT,yi,− 1 ,...,yi,−pi),m=2, 3

MQi,m=IT−Qi,m(Qi′,mQi,m)−^1 Qi′,m,m=2, 3

Xi,m=(yi,− 1 ,Qi,m),m=2, 3

MXi,m=IT−Xi,m(X′i,mXi,m)−^1 X′i,m,m=2, 3.

Then:

tiT,m(pi,φ ̃i)=

√

T−pi−m(y′i,− 1 MQi,myi)

(y′i,− 1 MQi,myi,− 1 )^1 /^2 (yi′MXi,myi)^1 /^2

,m=2, 3.

As before,m=2 refers to the specification with a constant but without a linear
trend, whilem=3 includes both a constant and a linear trend.
IPStis based on the cross-sectional average of the correctedt-statistics, that is,

IPSt,m=

√

N{tm−N^1

∑N

i= 1

E(tiT,m(pi,0)|ρi= 0 )}

√

1

N

∑N

i= 1

var(tiT,m(pi,0)|ρi= 0 )}

⇒N(0, 1), m=2, 3,

where:

tm=

1

N

∑N

i= 1

tiT,m(pi,φ ̃i),

andE(tiT,m(pi,0)|ρi= 0 )and var(tiT,m(pi,0)|ρi= 0 )are the mean and variance
of the Dickey–Fuller statistic respectively for finiteTand depend on the nuisance
parametersφ ̃i.AsT→∞, this dependence disappears andE(tiT,m(pi,0)|ρi= 0 )
and var(tiT,m(pi,0)|ρi= 0 )converge to the mean and variance of the Dickey–Fuller
density corresponding to the model estimated (with intercept or with intercept
and linear trend).
IPS tabulate these so-called correction terms for a set of values forTandpi=p
for both specifications of the deterministic terms. Use of these correction terms (for