Palgrave Handbook of Econometrics: Applied Econometrics

690 Panel Methods to Test for Unit Roots and Cointegration

y∗i=Miyi. Normalizing,
f ̃′f ̃
T− 1 =I, we have the estimate of the loading matrix:

 ̃=(π ̃ 1 ,...,π ̃N)′= y

∗′f ̃

T− 1

,

and:
̃zi,t=yi∗,t−f ̃t′π ̃i.

Finally:

̃ei,t=

∑t

s= 2

̃zi,s,

which can now be tested for a unit root, via ADF regressions. This is, in effect, the
test for the null of no cointegration, once the common factor structure and the
structurally unstable deterministic processes have been accounted for.
Thus, for each uniti, we estimate the regressions:

 ̃emi,t(λi)=αi,0 ̃eim,t− 1 (λi)+

∑k

j= 1

αi,j ̃emi,t−j(λi)+εi,t,m=c,τ,γ,

and test the null hypothesisH 0 :αi,0=0 either unit by unit or by constructing a
pooled panel test similar in spirit to LLC.
Three sub-cases,m=c,τ,γ, are considered by Banerjee and Carrion-i-Silvestre
(2007), which index thet-teststαmi,0(λi).m=cdenotes models which do not include
a time trend and the structural change affects either the intercept and/or the coin-
tegrating vector,m=τstands for models with a trend where, as before, the break
affects the intercept and/or the cointegrating vector, andm=γstands for models
where a change in trend is allowed.
The dependence on the break dates (which had been suppressed earlier) is made
clear here. No added estimations are needed at this stage because the breaks are
assumed known. In the more general case, however, the tests will have to be based
on estimates of the break dates for each of these units, as discussed for the general-
ization of the Pedroni tests above, but with the further complication of accounting
for cross-sectional dependence. For the models considered, the following result has
been derived.

Theorem (Banerjee and Carrion-i-Silvestre, 2007, Theorem 2): Under the null hypothesis
that ρi=φi− 1 =0:

(a)tαci,0(λi)⇒

1

(^2 (Wi(^1 )^2 −^1 )

∫ 1

0 Wi(s)^2 ds

) 1 / 2

(b)tατi,0(λi)⇒−^12

(∫ 1

0 Vi(s)

(^2) ds)−^1 /^2
whereVi(s)=Wi(s)−sWi( 1 )
(c)tαγi,0(λi)⇒−^12
(
λ^2
∫ 1
0 Vi(b^1 )
(^2) db
1 +
(
1 −λ^2
))∫ 1
0 Vi(b^2 )
(^2) db
2
whereVi(bj) =Wi(bj)−bjWi( 1 ),j =1, 2 are two independent detrended
Brownian motion processes.