Palgrave Handbook of Econometrics: Applied Econometrics

700 Panel Methods to Test for Unit Roots and Cointegration

Furthermore, group mean FM-OLS estimation is considered in Pedroni (2000),
with the estimator in its un-normalized form simply being given by the cross-
sectional average of the individual FM-OLS estimators:

βˆFMG =^1

N

∑N

i= 1

⎛

⎜

⎝

⎛

⎝

∑T

t= 1

x ̃i,tx ̃′i,t

⎞

⎠

− 1 ⎛

⎝

∑T

t= 1

̃xi,ty ̃i,t−

(

%ˆ+uv,i

)′

⎞

⎠

⎞

⎟

⎠.

Dynamic OLS

We now turn to dynamic OLS estimation of the cointegrating relationship as dis-
cussed in Kao and Chiang (2000) and Mark and Sul (2003). The idea of D-OLS
estimation is to correct for the correlation betweenvi,tandui,t(see equations

(13.18′)–(13.22′)) by including leads and lags ofxi,tas additional regressors in
the cointegrating regression. As in the time series case (in general), the number of
leads and lags has to be increased with the (time dimension of the) sample size at a
suitable rate to induce the noise process in the lead and lag augmented regression
andvi,tto be uncorrelated asymptotically. Thus, considering again the casem=2,
let the augmented cointegrating regression be given by:

y ̃i,t=x ̃

′

i,tβ+

∑pi

j=−pi

x ̃

′

i,t−jγi,j+u

∗

it

=x ̃

′

i,tβ+Z ̃

′

i,tγi+u

∗

it,

where the last equation defines Z ̃i,tandγi. The pooled D-OLS estimator for
β is then obtained from OLS estimation of the above equations. LetQ ̃i,t =
[
̃x

′

i,t,0

′

,...,0

′

,Z ̃

′

i,t,0

′

,...,0

′]

∈R

2 dim(x)

(

1 +

∑N

i= 1 pi

)
, where the variablesZ ̃i,tare at
theith position in the second block of the regressors. Using this notation we arrive
at:
⎡
⎢⎢
⎢⎢
⎣

βˆD

γˆ 1

..

.

γˆN

⎤

⎥⎥

⎥⎥

⎦

=

⎛

⎝

∑N

i= 1

∑T

t= 1

Q ̃i,tQ ̃

′

i,t

⎞

⎠

− 1 ⎛

⎝

∑N

i= 1

∑T

t= 1

Q ̃i,ty ̃i,t

⎞

⎠.

Mark and Sul (2003) obtain the asymptotic distribution ofβˆD, which has a sand-

wich type limit covariance matrix. Denote this asV = lim
N→∞

1

N

∑N

i= 1 ω

2
u.v,iv,i
(again assumed to exist), then it holds that:

N^1 /^2 T

(

βˆD−β

)

⇒N

(

0, 6−v^1 V ̄−v^1

)

.

Kao and Chiang (2000) discuss a normalized version of the D-OLS estimator that
corresponds toβˆFM^0. This estimator,βˆD^0 say, is obtained when, in the above discus-

sion of the D-OLS estimator,y ̃i,tandx ̃i,tare replaced byy ̃i^0 ,tandx ̃^0 i,t. These changes