Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 661

Choi (2006a)
The final specialization discussed here, due to Choi (2006a), is based on a two-way
error component model. Consider the model given by:

yi,t=β 0 +xi,t

xi,t=μi+λt+ui,t

ui,t=

∑pi

j= 1

αijui,t−j+εi,t.

Theμi,λtandεi,tprocesses (which are uncorrelated with each other) are assumed
to have the following structures:

E(μi)= 0 ∀i,E(μ^2 i)=σμ^2 <∞∀i,E(μiμj)= 0 ∀i=j

E(λt)= 0 ∀t,E(λtλs)=σλ(|t−s|)<∞

εi,t∼i.i.d.(0,σε^2 ), εi,tis independent ofεj,s∀i=j,s=t.

The test therefore consists of purging thexi,tprocess of its “common” time effects,
given byλt, its individual specific effects, given byμi, and testing for a unit root
inuˆi,t. The null and alternative hypotheses have the form:

H 0 :

∑pi

j= 1

αij= 1 ∀i; HA:

∑pi

j= 1

αij<1 for 0<N 1 <Nunits.

As earlier, the fraction of the units with no unit roots is required to fulfill the
following property under the alternative:

lim

N→∞

N 1

N

=k>0.

Consider testing the demeaned and detrended residualsuˆi,t, for each uniti, using
ADF tests (and critical values) and suppose the correspondingp-value of the test
for each unit is given bypi.
Choi (2006a) proposes the use of three different group mean tests based on the
Fisher principle (generalizing his earlier work on Fisher tests for cross-sectionally
independent panels):

CP=−√^1

N

∑N

i= 1

(log(pi)+ 1 )

CZ=√^1

N

∑N

i= 1

(#−^1 (pi))

CL∗=√^1

π^2 N 3

log

(

pi

1 −pi

)

.

Here#denotes the distribution function of the standard normal distribution.
Under the null, all three statistics tend toN(0, 1),T,N →∞. Under the alter-
native,CZ→∞while the remaining two statistics,CP,CZ→−∞,T,N→∞. This
framework can, of course, be generalized to incorporate a trend in the model.