Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 667

where:

λi,k=

Tbi,k

T

,k=1,...,mi+ 1 , which are fractions that remain constant asT→∞

λi=(λi,0,.λi,k,...,λi,mi+ 1 )′, withλi,0=0,λi.mi+ 1 =1,i=1, 2,...,N,

and:
Vib,k=Wi,k(b)−bWi,k( 1 ),

are Brownian bridges independent acrossiandk.
Note the dependence of the statistics on the break fractions. Thus, denoting
λ=(λ 1 ,...,λN)′, we have:

Z=

√

N

MSB−ξ

ζ

⇒N(0, 1)

MSB=

1

N

∑N

i= 1

MSB(i,λi)

ξi=

1

6

m∑i+ 1

k= 1

(λi,k−λi,k− 1 )^2

ζi^2 =

1

45

m∑i+ 1

k= 1

(λi,k−λi,k− 1 )^4

ξ=

1

N

∑N

i= 1

ξi

ζ^2 =

1

N

∑N

i= 1

ζi^2.

The Fisher orp-value versions of these testsàlaChoi (2006a), referred to asBCN,
andàlaMaddala and Wu (1999), referred to asBCχ 2 , can similarly be constructed,
as long as the break fractions are known. This is because thep-values for the
individual tests will depend on the break fractions in the presence of breaks in
trend.

13.2.3.2 Break dates unknown

The results and methods discussed above go through as before for Model 2 (note
that for Model 1 knowledge of the break fractions is not a relevant consideration)
provided consistent estimates of the break fractions can be obtained. From (13.15),
defining the composite errorni,t=F′tπi+e∗i,t, and assuming without loss of
generality thatFhas zero mean, we can write the model as:

yi,t=δi+

∑mi

k= 1

γi,kDUi,k,t+ni,t.