Katarina Juselius 369Thus, unless we restrictα⊥ 2 ρ 2 = 0 the model will allow for cubic trends in the
data. TheI( 2 )procedure in CATS (Cointegration Analysis of Time Series) in RATS
(Regression Analysis of Time Series) (Dennis, Johansen and Juselius, 2005) imposes
this restriction. The effect of cumulating the linear trend term once is given by:
C 1∑tj= 1μ 1 j=∑tj= 1(ω 0 α′+ω 1 α′⊥ 1 +ω 2 α′⊥ 2 )(αρ 0 +α⊥ 1 ρ 1 +α⊥ 2 ρ 2 )j=∑tj= 1(ω 0 α′αρ 0
︸︷︷︸
= 0+ω 1 α′⊥ 1 α⊥ 1 ρ 1
︸ ︷︷︸
= 0+ω 2 α′⊥ 2 α⊥ 2 ρ 2
︸ ︷︷︸
= 0)j. (8.30)It appears that all threeC 1 components of the linear trend will generate quadratic
trends in the data. Based on (8.29), we already know thatα⊥ 2 ρ 2 = 0. Unless we
are willing to accept linear trends inα′⊥ 1 xt,^8 we should also restrictα⊥ 1 ρ 1 = 0.
This leaves us with theαcomponent ofC 1 , which cannot be set to zero, because
αρ 0 = 0 is needed to allow for a linear trend inβ′xt. The problem is that a lin-
ear trend in a polynomially cointegrating relation, unless adequately restricted,
generates a quadratic trend inxt. However, this can be solved by noticing that
α⊥ 2 γ 2 = 0 in (8.27) also generates a quadratic trend inxt, so that by restricting
ω 0 α′αρ 0 =−β⊥ 2 α′⊥ 2 α⊥ 2 γ 2 , the two trend components cancel and there will be
no quadratic trends in the data. The trend-stationary polynomially cointegrated
relation in Kongsted, Rahbek and Jørgensen (1999) was estimated subject to this
constraint.
To summarize: to avoid quadratic and cubic trends in theI( 2 )model we need to
impose the following restrictions:ρ 1 =ρ 2 = 0 andω 0 α′αρ 0 =−β⊥ 2 α′⊥ 2 α⊥ 2 γ 2 ,as
well asρ 11 =ρ 21 = 0 andω 0 α′αρ 01 =−β⊥ 2 α′⊥ 2 α⊥ 2 γ 21 to avoid broken quadratic
and cubic trends.
8.7 Estimation in theI( 2 )model
Johansen (1995) provided the solution to the two-step estimator and Johansen
(1997) to the full maximum likelihood (ML) estimator. Even though the two-stage
procedure gives asymptotically efficient ML estimates (Paruolo, 2000), the small
sample properties of the ML estimates are generally superior (Nielsen and Rahbek,
2007), and all subsequent results are based on the ML procedure.
8.7.1 The ML procedure
Section 8.2 showed that there is an important difference between the first- and
second-rank conditions. The former is formulated as a reduced rank condition
directly on, whereas the latter is on a transformed. This is the basic reason
why the ML estimation procedure needs a different parameterization than the one
in (8.3).