Palgrave Handbook of Econometrics: Applied Econometrics

704 Panel Methods to Test for Unit Roots and Cointegration

spaces, for example, normalized asβˆi=

[

Ik,βˆ

′

i,2

]′
, which are in any case computed
in the derivation of the test statistic.

13.3.4 Breitung (2005)

Breitung (2005) proposes a two-step estimation and test procedure that extends the
Ahn and Reinsel (1990) and Engle and Yoo (1991) approach from the time series
to the panel case.
Breitung considers the homogeneous cointegration case where, however, only
the cointegrating spaces are assumed to be identical for all cross-section members.
In the first step of his procedure the parameters are estimated individual specifically
(applying the method outlined in Johansen, 1995). This includes the first-step
estimates ofβi. In the second step the common cointegrating spaceβis estimated
in a pooled fashion.
For simplicity we describe the method here for the VAR(1) model excluding any
deterministic component. In the general case lagged differences as well as poten-
tially restricted deterministic components are treated in the usual way by being
concentrated out at the beginning of the procedure. Thus, consider the following:

Yi,t=αiβ

′

Yi,t− 1 +wi,t. (13.32)

Next, defineTi=

(

α

′

i

− 1

i αi

)− 1

α

′

i

− 1
i and pre-multiply (13.32) with this quantity
to obtain:
(
α

′

i

− 1

i αi

)− 1

α

′

i

− 1

i Yi,t=β

′

Yi,t− 1 +

(

α

′

i

− 1

i αi

)− 1

α

′

i

− 1

i wi,t

TiYi,t=β

′

Yi,t− 1 +Tiwi,t

Yi+,t=β

′

Yi,t− 1 +w+i,t, (13.33)

where the last equation defines the variables with superscript+. Note also that

E[w+i,t(w+i,t)

′

]=

(

α

′

i

− 1

i αi

)− 1

. Next use the normalizationβ=

[

Ik,β

′

2

]′

and partition

Yit=

[

(Yi^1 ,t)′,(Yi^2 ,t)′

]′

withYi^1 ,t∈RkandYi^2 ,t∈Rm−k. Using this notation the above

equation (13.33) can be rewritten as:

Yi+,t−Yi^1 ,t− 1 =β

′

2 Y

2

i,t− 1 +w

+

i,t. (13.34)

Breitung suggests estimating (13.34) by pooled OLS using the estimatesTˆi =
(
αˆ

′

iˆ

− 1

i αˆi

)− 1

αˆ

′

iˆ

− 1
i based on the individual specific Johansen estimates. Note that,
given that the covariance structure of the errors in (13.34) is known and an esti-
mate is readily available, pooled feasible GLS estimation of (13.34) can also be
performed.
Breitung’s estimation procedure stops here. However, an iterative estimator based
on the above procedure is easily conceived. With the estimatedβˆ 2 , all individual-
specific parameters in (13.32) can be re-estimated. Since we have chosen the VAR(1)