Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 691

Moreover, if we assume for the moment that there is only one factor, then the
estimated factorF ̃tcan also be tested for a unit root using an ADF regression of the
form:

F ̃mt(λi)=δ 0 F ̃tm− 1 (λi)+

∑k

j= 1

δjF ̃tm−j(λi)+ut.

Then, for cases where there is no break in trend:

(d) tδ 0 ⇒

∫ 1

(^0 Wd(s)dWd(s)

∫ 1

0 Wd(s)^2 ds

) 1 / 2

whereWd(s)denotes a detrended Brownian motion. However, allowing for a

change in trend leads to dependence on the break fraction, such that:

(e) tδ 0 (λ)⇒

∫ 1

(^0 Wd(s,λ)dWd(s,λ)

∫ 1

0 Wd(s,λ)^2 ds

) 1 / 2.

It is key to note the following features of the results:

(i) The densities derived above show no dependence on the stochastic regressors
(that is, thexi,tprocesses). This follows from assuming orthogonality of the
stochastic regressors to the factors and exogeneity with respect to the idiosyn-
cratic errors. It may be shown that this implies that the above results also hold
in cases where breaks in the cointegrating vector occur.
(ii) The limiting distributions, as long as a change in trend is not involved, do not
depend on the break dates. This implies that, in principle, multiple changes
in constants and slopes of the cointegrating vectors are allowed to occur.
Heterogeneous break dates are allowed, and the issue of break dates being
known or unknown is no longer relevant.
(iii) The situation changes substantially when a break in trend is allowed. The break
fraction is important and thus must be estimated if not known. Moreover,
recalling the discussion above, for the factor structure to be preserved, we need
the breaks to occur at the same time across the units, so that the projection
matrixMis not indexed byi. Thus trend breaks, if they occur, must have the
sameλin all the units
(iv) Finally, for the case of several common factors, that is,r>1, variations of the
MQtests proposed by Bai and Ng (2004) can be used and are subject to the
same remarks as above. Models that do not allow for changes in trend show no
dependence either on the stochastic regressors or, more importantly, on the
timing of the breaks. Models that allow for a change in trend lead to statistics
that depend upon the common break date.

The pooled test statistics are given by:

N−^1 /^2 ZtcNT,N−^1 /^2 ZτtNTandN−^1 /^2 ZtγNT,