Palgrave Handbook of Econometrics: Applied Econometrics

682 Panel Methods to Test for Unit Roots and Cointegration

pooledρ-test: N^1 /^2

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^1

∑T

t= 2

uˆi,t− 1 uˆi,t−ψi

)

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^2

∑T

t= 2

uˆ^2 i,t− 1

)

pooled t-test: N^1 /^2

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^1

∑T

t= 2

uˆi,t− 1 uˆi,t−ψi

)

ωˆN,T

(

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^2

∑T

t= 2

uˆ^2 i,t− 1

)) 1 / 2.

(ii) Pooled tests (parametric corrections)

In order to compute the parametrically corrected versions of thet-test, a similar

device to that discussed previously in the construction of the LLC test is used.

Two auxiliary regressions are estimated:

uˆi,t=

∑Ki

k= 1

γ1,ikuˆi,t−k+ζ1,i,t

uˆi,t− 1 =

∑Ki

k= 1

γ2,ikuˆi,t−k+ζ2,i,t,

where the lag-length selection (ofKi)may be undertaken using automatic

selection criteria such as AIC. Next,ζˆ1,i,tis regressed onζˆ2,i,t:

ζˆ1,i,t=ρiζˆ2,i,t+θi,t,

andσˆNT^2 =NT^1

∑N

i= 1

∑T

t=Ki+ 2 θˆ

2

i,tis computed. The variance of the estimated

residualsθˆi,t, denotedσˆθ^2 i, needed for the computation of the group mean tests

is also computed. The parametrically corrected pooledt-test is then:

pooled t-test–parametrically corrected:

N^1 /^2

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^1

∑T

t=Ki+ 2

ζˆ1,i,tζˆ2,i,t

)

σˆN,T

(

N−^1

∑N

i= 1

ωˆ−u.^2 ν,i

(

T−^2

∑T

t=Ki+ 2

ζˆ^2

2,i,t

)) 1 / 2.

The group mean tests are defined as follows: