Palgrave Handbook of Econometrics: Applied Econometrics

Katarina Juselius 365

8.6 Representing theI( 2 )model

8.6.1 The basic structure

As discussed in section 8.2, formulation (8.3) is convenient when data areI( 2 ):

^2 xt=

xt− 1 +xt− 1 +μ 0 +μ 01 Ds,91.1,t+μ 1 t+μ 11 t91.1+pDp,t+εt, (8.16)

where the deterministic components are defined in section 8.5.1. Similar to the
I( 1 )model, we need to define the concentratedI( 2 )model:^6

R0,t=R1,t+R2,t+εt, (8.17)

whereR0,t,R1,t, andR2,tare defined by:

^2 x ̃t=bˆ 10 +bˆ 11 t+Bˆ 11 Ds,t+Bˆ 12 Dp,t+R0,t, (8.18)

x ̃t− 1 =bˆ 20 +bˆ 21 t+Bˆ 21 Ds,t+Bˆ 22 Dp,t+R1,t, (8.19)

x ̃t− 1 =bˆ 30 +bˆ 31 t+Bˆ 31 Ds,t+Bˆ 32 Dp,t+R2,t, (8.20)

andx ̃tindicates thatxthas been augmented with some deterministic components
such as trend, constant, and shift dummy variables. The matricesandare
subject to the two reduced rank restrictions,=α′β, whereα,βarep×r, and
α′⊥β⊥=ξη′, whereξ,ηare (p−r)×s 1. The model in (8.16) contains an unrestricted
constant with a shift, a broken trend and a few impulse dummies that will have to
be adequately restricted to avoid undesirable effects, as discussed in section 8.6.2.
The moving average representation of theI( 2 )model (Johansen, 1992, 1995,
1997) with unrestricted deterministic components is given by:

xt=C 2

∑t

j= 1

∑j

i= 1

(εi+μ 0 +μ 1 i+μ 01 Ds,91.1,i+pDp,i)

+C 1

∑t

j= 1

(εj+μ 0 +μ 1 j+μ 01 Ds,91.1,j+pDp,j) (8.21)

+C∗(L)(εt+μ 0 +μ 1 t+μ 01 Ds,91.1,t+pDp,t)+A+Bt,

whereAandBare functions of the initial valuesx 0 ,x− 1 ,x− 2 , and the coefficient
matrices satisfy:

C 2 =β⊥ 2 (α′⊥ 2 !β⊥ 2 )−^1 α′⊥ 2 ,

β′C 1 =−α′C 2 , β′⊥ 1 C 1 =−α′⊥ 1 (I−C 2 ),

=βα′+I− 1 (8.22)

where the notationα=α(α′α)−^1 is used throughout the chapter. To facilitate the
interpretation of theI( 2 )stochastic trends and how they load into the variables, it
is useful to letβ ̃⊥ 2 =β⊥ 2 (α′⊥ 2 β⊥ 2 )−^1 , so that:

C 2 =β ̃⊥ 2 α′⊥ 2. (8.23)