Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 699

in (13.18′), which puts the homogeneity restriction on the cointegrating rela-
tionship in place. Next, defining the endogeneity corrected variable ̃y+i,t=y ̃i,t−

ˆuv,iˆ−v,^1 ix ̃i,tleads to the following pooled FM-OLS estimator:

βˆFM=

⎛

⎝

∑N

i= 1

∑T

t= 1

x ̃i,t ̃x

′

i,t

⎞

⎠

− 1 ⎛

⎝

∑N

i= 1

∑T

t= 1

(

x ̃i,ty ̃i+,t−

(

%ˆ+uv,i

)′)

⎞

⎠,

where%+uv,i=%ˆuv,i−ˆuv,iˆ−v,^1 i%ˆv,i. Phillips and Moon (1999) use in their for-

mulation of the FM-OLS estimator averaged correction factors, for example,ˆv=
1
N

∑N
i= 1 ˆv,iand similarly for the other required matrices. The limiting distribution
of the FM-OLS estimator (see, for example, Theorem 9 of Phillips and Moon, 1999)
is given by:

N^1 /^2 T

(

βˆFM−β

)

⇒N

(

0, 6ω^2 u.v−v^1

)

,

withωu^2 .v= lim
N→∞

1

N

∑N

i= 1 ω

2

u.v,iandv=Nlim→∞

1

N

∑N

i= 1 v,i, with these limits (in

most papers implicitly) assumed to exist.^29 For the casem=0 (without determin-
istic components), the multiplicative factor 6 in the variance term of the limiting
distribution has to be replaced by 2. Also note that the limiting covariance matrix
is composed of cross-sectional averages.
Standard, up to the factor 2 or 6 depending upon the deterministic specifica-
tion considered, normally distributed pooled FM-OLS estimators are also easily
constructed. These are popular due to their implementation in freely available
software. Define the following:

̃yi^0 ,t=ˆω−u.^1 v,iy ̃+i,t−

[(

ωˆ−u.^1 v,iIdim(x)−ˆ−v,i^1 /^2

)

x ̃i,t

]′

βˆ

̃xi^0 ,t=ˆ−v,^1 i/^2 x ̃i,t

%ˆ^0 uv,i=ˆω−u.^1 v,i%ˆ+uv,iˆ−^1 /^2

v,i ,

whereIdim(x)denotes the identity matrix with dimensional equal to the number
of regressorsxi,t. andβˆdenotes the LSDV estimator. Then, the normalized FM-OLS
estimator is given by:

βˆFM^0 =

⎛

⎝

∑N

i= 1

∑T

t= 1

x ̃^0 i,t ̃x^0

′

i,t

⎞

⎠

− 1 ⎛

⎝

∑N

i= 1

∑T

t= 1

(

x ̃^0 i,ty ̃i^0 ,t−

(

%ˆ^0 uv,i

)′)

⎞

⎠,

for which it holds thatN^1 /^2 T

(

βˆFM^0 −β

)

⇒N

(

0, 6Idim(x)

)
, where again the factor
6 has to be replaced by the factor 2 in the case of no deterministic components.