654 Panel Methods to Test for Unit Roots and Cointegration
(iii) For eachi,
εi,t∼i.i.d.(0,σε^2 i),E|εi,t|^8 ≤A,∑∞j= 0j||Hi,j||<A,ωi^2 =Hi( 1 )^2 σε^2 i>0;εi,tare independent overi.^15
(iv) The errorsεj,t,ηs, and the loadingsπiform three mutually independent groups
for all (j,t,s,i).^16
(v)E||F 0 || ≤A, and for alli=1,2,...,N,E|ei,0|≤A.
In the assumptionsAis taken to be a positive number not depending on eitherT
orN. The notation||B|| =trace(B′B)^1 /^2.
Both the factors and the idiosyncratic components can be integrated or station-
ary, so that short- and long-run dependence can be modeled both via the common
factors and also the idiosyncratic components. However, in our discussion we
assume that in (13.7) the idiosyncratic terms are taken to be independent acrossi,
which is slightly stronger than the assumptions necessary for applicability of the
Bai and Ng (2004) methods.
The heart of the unit root analysis consists of making the decomposition
(between common factors and idiosyncratic terms) and then testing each of these
components for a unit root. Thus, returning to the set-up described by (13.5), let:
yi,t=μi+πi′Ft+ei,t,whereFtis anr×1 vector of “common factors.” The model can be rewritten in
first differences:
yi,t=π′ift+zi,t, i=1, 2,...,N;t=2, 3,...,T,where:
ft=Ft,t=2, 3,...,T
zi,t=ei,t,i=1, 2,...,N;t=2, 3,...,T.
Next, define:
Y=(y 1 ,y 2 ,...,yN),
as theT×Nmatrix of all observations, where:
yi=(yi,1,yi,2,...,yi,T)′,and:
y ̃=Y=(y 1 ,...,yN),
is the corresponding(T− 1 )×Nmatrix of first differences. The principal component
estimatorfˆoff=(f 2 ,f 3 ,...,fT)′is
√
T−1 times thereigenvectors corresponding
to therlargest eigenvalues of the(T− 1 )×(T− 1 )matrixy ̃y ̃′and the estimated
factor loading matrix given byπˆ=(πˆ 1 ,...,πˆN)′is obtained from the relationship