Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 659

rewritten in the form given by (13.5)–(13.7) above):

yit=μi+y^0 i,t

y^0 i,t=φiyi^0 ,t− 1 +ui,t

ui,t=πi′ft+ei,t

i=1, 2,...,N;t=1, 2,...,T.

Note thatftis anr-dimensional vector of common factors, taken here to be
stationary, see assumptions (ii) and (v) below, wherermay be taken to be known.^17
Key assumptions within this framework include the following:

(i)ei,t=

∑∞

j= 0 di,jεi,t−j, whereεi,tare i.i.d. (0,1) acrossiand overt, have finite

eighth moment, infi

∑∞

j= 0 di,j>0 anddj=supi|di,j|,

∑∞

j= 0 j

md

j<Mfor some

m>1;

(ii)ft=

∑∞

j= 0 cjηt−j, wherecjarer×rmatrices of real numbers and ther×1 vectors

ηt=(η1,t,...,ηj,t,...,ηr,t)′are i.i.d.(0,Ir), so thatηj,tis i.i.d acrossjand over

t. It is also assumed that

∑∞

j= 0 j

m||c
j||<M, for somem>1;
(iii)εi,tandηi,sare independent;

(iv) asN→∞,N^1

∑N

i= 1 πiπ

′

i→>0;

(v) asT→∞,^1 T

∑N

i= 1 ftf

′

t→f>0.

The unit root in theyi,tprocess comes solely fromφibeing equal to one. This is
a key restriction from the overall Bai and Ng (2004) framework (which allows for
integrated factors and integrated idiosyncratic components).
Following from above, the unit root null hypothesis ofH 0 :φi= 1 ∀iis therefore
tested against the heterogeneous alternative ofHA:φi<1 for some of the units, as
long as the number of these units remains a positive fraction of the total number
of units asN→∞. The vectorπidetermines the loadings of the factors into theith
unit and, ifr=1, the system simplifies to the DGP considered by Pesaran (2007).^18
The procedure consists of first computing the pooled OLS estimatorφˆPooled(ofφ),
given by settingφi=φ∀iand calculating the residuals from the pooled regression
asuˆi,t=yi,t−ˆφPooledyi,t− 1. Extracting the factors from theuˆi,tseries is the second
step.
Next, we establish some notation:

%ˆ=(πˆ 1 ,πˆ 2 ,...,πˆN)′, whereπˆi,i=1, 2,...,N, are the estimated factor loadings

Qˆˆ

%=IN−

%(ˆ%ˆ′%)ˆ−^1 %ˆ′

yi,− 1 =(yi,0,yi,1,...,yi,T− 1 )′

yi=(yi,1,...,yi,T)′

Y=(y 1 ,y 2 ,...,yN)