It is now easy to see that the C^ matrix has a similar reduced rank representation as C1
in the I(1) model, so that it is straightforward to interpret o' 2 Σ2 εi as a measure of
the s2 second order stochastic trends which load into the variables xt with the weights
Ъ
β±2 ■
From (9) we note that the C1 matrix in the I(2) model cannot be given a simple
decomposition as it depends on both the C2 matrix and the other model parameters
in a complex way. Johansen (2005) derived an analytical expression for C1, essentially
showing that:
Ci = ω⅛o' + wɪɑɪɪ + ω∙2θ.' 2 (ɪɪ)
where ω⅛ are complicated functions of the parameters of the model (not to be repro-
duced here).
3 The ML procedure
The full ML procedure derived in Johansen ^997) exploits the fact that the I(2)
model contains r + sɪ cointegration relations of which the r relations, β'xt — I(1),
can become stationary by polynomial cointegration, β'xt + 0∆xt — I(0), and the
sɪ relations, ββ1xt — I(1), can become stationary by differencing, ββ1∆xt — I(0)
Thus, τ = (β,βxi) define the r + sɪ = p — s2 directions in which the process is
cointegrated from I(2) to I(1) This means that the space spanned by τ = (β, βɪɪ) can
be determined by solving just one reduced rank regression, after which the vector space
can be separated into β and βxr However, this necessitates a re-parametrization of
the I(2) model. The following parametrization (here extended with the deterministic
components discussed above) was suggested by Johansen ^997):
∆2xi = o(p'T'Tt-ɪ + 0'∆.T∕ ∣ ) + ω'T'∆.t-, ∣ + ΦpDpjt + ΦfrDtr,t + εt,
εt - iɪd^ Np(0, Ω ), t = 1,^,T
^2)
where p is a (r + sɪ) × r matrix which picks out the r cointegration vectors β'xt (so
that p'τ' = β'), β' = — (a'Ω la) 1 o'Ω ∣Γ, ω' = Ωo (o' Ωo ) ∣ (o' Γ7, ξ), p'T' =
[β',P0,P0L τ = [≠',7o,7oJ, τt = [xt,t,t9id and ^t = ^xt, 1,Ds91 : 1], where t9i.i
is a linear trend starting in ɪ99ɪ:ɪand Ds9F1 is a step dummy starting in ɪ99ɪ:ɪ.
The relations T'∆xt define the p — s2 stationary relations between the growth rates,
of which r correspond β'∆xt + p0 + p01Ds9F1 and sɪ to ββ1∆xt + T0 + 701Ds91 :1 In
some cases they might be given an interpretation as medium run steady-state relations.
Based on an iterative estimation algorithm, o and β are estimated subject to the
reduced rank restriction(s) (7) on the Γ matrix. This is the reason why the estimates
of o and β based on the ML procedure can differ to some degree from the estimates
based on the I(1) model.
The FIML estimates of τ = (β,βxi) are obtained using an iterative procedure
which at each step delivers the solution of just one reduced rank problem. In this case
the eigenvectors are the estimates of the CI(2,1) relations, τ'xt, among the variables
xt, i.e^ they give a decomposition of the vector xt into the r + sɪ directions τ = (β, βɪɪ)