in which the process is I(1) and the s2 directions τɪ = β±2 in which it is I(2). The ML
parameterization avoids the problem of quadratic trends by restricting the constant
term, the linear trend, and the step dummy to the various cointegration relations.
The matrix β in (12) does not make a distinction between stationary and nonsta-
tionary components in ∆xt. For example, when xt contains variables which are I(2) as
well as I(1), then some of the differenced variables picked up by ψ will be I(0). As
the latter do not contain any stochastic I(1) trends, they are by definition excludable
from the polynomially cointegrated relations. The idea behind the parametrization in
Paruolo and Rahbek (1999) was to express the polynomially cointegrated relations
exclusively in terms of the I(1) differenced variables by noticing that
≠' ∆.x; I = ≠' (TT ' + T±T '' )∆.x; I
so that (12) can be reformulated as:
∆Λ⅛ = a {[β',p0ι,p0]}
ʃ (0)
xt-ι
t91:1
t
+ [^,7 01,7 o]
∆xt-1
^s91 : 1t-ι
const
+ζ
β , p0, Poi
β ±i,7o, 701
∆xt-1
^s91 : 1t-1
const
+ ^pDptt + Φtr Dtr,t + -O
(13)
εt - Ap(0, Ω),t =1,...,T
where δ' = ψτ±τɪ6 and ζ = aψ T + ω'. Note that this parameterization defines the
I(2) model directly in terms of stationary components.
4 Testing hypotheses in the I(2) model
We discuss in this section hypotheses on the parameters a, α±1, α±2, β, and τ, in he
maximum likelihood parametrization (12) written as
∆2xt = a(pτ 'xt-1 + ≠'τ±τ 2∆xt-1) + (α≠'τ + ω')τ '∆xm + ɛt,
ignoring deterministic terms. The general theory for likelihood ratio tests for such
hypotheses is given in Boswijk (2000) and Johansen (2006) and we here discuss the
interpretation of the hypotheses and apply the result that likelihood ratio statistics are
generally asymptotically distributed as χ2, except in a few cases, which we describe in
some more detail. In all cases the likelihood ratio tests are calculated by maximizing the
likelihood function, L, with the parameters restricted by the hypothesis and without
the restriction. The test statistic is — 2 log(maxrestr⅛ted L/ maxωrarestr⅛ted L). For each
case we give the degrees of freedom for the asymptotic χ2 distribution.
6Note that the definition of δ differes from the one used in Johansen (1997, 2006).
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