370 The Long Swings Puzzle
The full ML procedure exploits the fact that theI( 2 )model containsp−s 2 cointe-
gration relations,τ′xt, whereτ=(β,β⊥ 1 )definer+s 1 =p−s 2 directions in which
the process is cointegrated fromI( 2 )toI( 1 ). This means thatτcan be determined
by solving just one reduced rank regression, after which the vectorxtis decom-
posed into thep−s 2 directionsτ=(β,β⊥ 1 )in which the process isI( 1 ), and the
s 2 directionsτ⊥=β⊥ 2 in which it isI( 2 ).
Johansen (1997) does not make a distinction between stationary and non-
stationary components inxt. For example, whenxtcontains variables which
areI( 2 ), e.g., prices, as well asI( 1 ), e.g., nominal exchange rates, then some of the
differenced variables will beI( 0 ). As the latter do not contain any stochasticI( 1 )
trends, they are by definition redundant in the polynomially cointegrated rela-
tions. The idea behind the parameterization in Paruolo and Rahbek (1999) was to
express the polynomially cointegrated relations exclusively in terms of the differ-
ences of theI( 2 )variables. The model given below is based on the Paruolo and
Rahbek parameterization. As discussed in section 8.6, the (broken) trend has been
restricted to be proportional toα,and the constant and the shift dummy to be
proportional toζ.
^2 xt
︸︷︷︸
I( 0 )=α⎧
⎪⎪⎪
⎪⎪⎪
⎪⎨⎪⎪⎪
⎪⎪⎪
⎪⎩[
β′,ρ 0 ,ρ 01]⎡
⎢
⎣xt− 1
t
t91.1⎤
⎥
⎦︸ ︷︷ ︸
I( 1 )+[
δ′,γ 0 ,γ 01]⎡
⎢
⎣xt− 1
c
Ds91.1t− 1⎤
⎥
⎦︸ ︷︷ ︸
I( 1 )⎫
⎪⎪⎪
⎪⎪⎪
⎪⎬⎪⎪⎪
⎪⎪⎪
⎪⎭
︸ ︷︷ ︸
I( 0 )+ζ⎡
⎢⎢
⎢
⎣β′,ρ 0 ,ρ 01
β′⊥ 1 ,γ ̃ 0 ,γ ̃ 01
︸ ︷︷ ︸
τ⎤
⎥⎥
⎥
⎦⎡
⎢
⎣xt− 1
c
Ds91.1t− 1⎤
⎥
⎦︸ ︷︷ ︸
I( 0 )+pDp,t+εt,(8.31)whereεt∼Np( 0 ,),t=1,...,T,δ′=ψ′τ⊥τ′⊥withψ′=−(α′−^1 α)−^1 α′−^1 ,
ζ′=ψ′τ−α⊥(α′⊥α⊥)−^1 (α′⊥β,ξ)andξis defined in (8.5).
The relationsβ′x ̃t+δ ̃
′
[ x ̃t, withβ′ =[β′,ρ^0 ,ρ^01 ],x ̃t′ =[xt′,t,t91.1],δ ̃′ =
δ′,γ 0 ,γ 01
]
, andx ̃′t =[x′t,1,Ds91.1], definerstationary polynomially co-integrating relations, whereas the relationsτ′x ̃tdefinep−s 2 stationary relations
between the growth rates.
8.7.2 LinkingI( 1 )withI( 2 )
It is useful to see how the formulation (8.31) relates to the usual VAR formulation
(8.3). Relying on results in Johansen (1997), the levels and difference components