Anindya Banerjee and Martin Wagner 689the limiting distributions of the statistics do not depend on the stochastic regres-
sors as given in Theorem 2 of Banerjee and Carrion-i-Silvestre (2007). However, if,
for example,E(ei,t|νi,t)=x′i,tAi(L)+ξi,t, withAi(L)being a vector of lag and lead
polynomials andξi,t∼i.i.d.(0,ξ), the regressors are no longer strictly exogenous
and modifications must be introduced to account for the endogenous regressors.
The two theorems given below are for the case of exogenous regressors but a simple
modification of the testing procedure is suggested to allow for endogeneity.
Consider the case where the break dates are known, so that for the deterministic
components only the corresponding parameters need to be estimated.
Compute the first difference of (13.18) to give:
yi,t=Di,t+x′i,tδi,t+F′tπi+ei,t.Note that, depending on the specification of the deterministic breaks, the differ-
enced deterministic componentDi,tis a mixture of step functions (in the case of
a break in trend) and impulse dummies (when there is a break in intercept).
yi=(yi,2,...,yi,T)′ei=(ei,2,...,ei,T)′F=(F 2 ,...,FT)′((T− 1 )×r)matrix of differenced factors for all the units)πi=(πi,1,...,πi,r)′(r×1 vector of loadings of factors forith unit)xi=(xi,2,...,xi,T)′.Defining the projection matrixMi = (IT− 1 −xDi(xDi′xDi)−^1 xDi′), with
xDi,t=(1,Di,t,x′i,t)′andxDi =(xDi,2,...xDi,T)′, we have:
Miyi=MiFπi+Miei
=fπi+zi,wheref=MiFandzi=Miei. The projection matrixMiis sensitive to the speci-
fication of the deterministic terms, as indicated by the indexDin the definition of
this matrix; yetMiFcannot be sensitive toiif the common factor representation
is to be valid. This is a restriction of the framework used to model dependence, and
can be satisfied in two different ways – (a) either the deterministic components
do not matter, as in the case where differencing leads to impulse dummies, or (b)
where the timing of the breaks in the deterministic terms is the same across the
units. This is relevant in the case of trend breaks, where differencing leads to shifts
in intercept which are relevant for the derivation of the asymptotic distributions
of the statistics, that is, for Models 3 and 6.
Pre-multiplication by this projection matrix serves to isolate the factor compo-
nents, so that we can now proceed to extract the factors. The estimated factors,
denotedf ̃=(f ̃ 2 ,...,f ̃T)are given by
√
T−1 times thereigenvectors correspond-
ing to therlargest eigenvalues of the matrixy∗y∗′, wherey∗=(y 1 ∗,...,yN∗)′and