variables in differences. The intuition is that the differenced process also contains unit
roots when data are I(2).
There is, however, an important difference between the first and the second condi-
tion. The former is formulated as a reduced rank condition directly on ∏, whereas the
latter is on a transformed Γ. Below we shall show that this is the basic reason why the
ML estimation procedure needs a different parameterization than the one in (1) .
The intuition behind (7) can be seen by pre-multiplying (1) with aβ. This makes
the levels component aβ'xt~1 disappear and reduces the model to a (p — r)-dimensional
system of equations in first- and second order differences. In this system the hypothesis
of reduced rank of the matrix o, Γβ' is tested in the usual way. Thus, the second
condition is similar to the first except that the second reduced rank is formulated on
the p — r common driving trends, rather than on the p variables.
Using (7) it is possible to decompose о and βɪ into the I(1) and I(2) directions
α = [αχ1,αχ2] and βɪ = [βx1,β±2] . The matrices ɑɪɪ and β11 of dimension p × sɪ,
and α 2 and β±2 of p × s2 are defined by ɑɪɪ = α±ξ, βɪ 1 = β1η, a 2 = aχξx and
β 2 = β±L±, where ζ±,η± are the orthogonal complements of ξ and η, respectively5
and a = о ('o, о ) 1 denotes a shorthand notation used all through the chapter.
The moving average representation of the I(2) model was derived in Johansen
(1992). The baseline VAR model (1) contains a constant, a trend and several dummy
variables that will have to be restricted in certain ways to avoid undesirable effects.
Without such restrictions the MA model can be given in its completely unrestricted
form:
t j
Xt = C2 ∑2 ∑(εi + Lo + L1i + ΦsDs,i + ΦpDpji + Φ/.,.Dtr,i)
j=1 i=1
t
(8)
+ C1 ^ (£j + Lo + L1J + ®sDs,j + ΦpDp,j + $trDtr,j)
j=1
+ C *(L)(£t + Lo + L1t + ®sDs,t + ®pDp,t + Φtr Dtr,t) + A + Bt
where A and B are functions of the initial values Xo, x~1, ...,x~k+1, and the coefficient
matrices satisfy:
C2 = β n(⅛2Φβ 2 2,
β'C1 = —U'ΓC2, β!1 C1 = — ⅛1(I — ΦC2),
(9)
Φ = ΓβαT + r1
To facilitate the interpretation of the I(2) trends and how they load into the variables,
j £D £D ( 1 1
we denote ɔ' 2 = βx2(α,h2Φβ±2) 1, so that
_ ~ , .
C2 = β 2 ,. (10)
5Note that the matrices a±1, a±2, βɪo and β±2 are called a1, a2. β 1 and β2 in the many papers
on I(2) by Johansen. The reason why we deviate here from the simpler notation is that we need
to distinguish between different β and a vectors in the empirical analysis and, hence, use the latter
notation for this purpose.