Testing Hypotheses in an I(2) Model with Applications to the Persistent Long Swings in the Dmk/$ Rate

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 x_t 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'x_t — I(1),
can become stationary by polynomial cointegration, β'x_t + 0∆x_t — I(0), and the
sɪ relations, β^β₁x_t — I(1), can become stationary by differencing, β^β₁∆x_t — I(0)
Thus, τ = (β,β_xi) 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 β_xr 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):

∆²x_i = o(p'T'Tt-ɪ + 0'∆.T∕ ∣ ) + ω'T'∆.t-, ∣ + Φ_pD_pjt + Φ_frD_tr,_t + ε_t,
ε_t - iɪd^ N_p(0, Ω ), t = 1,^,T

where p is a (r + sɪ) × r matrix which picks out the r cointegration vectors β'x_t (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^{, τ}t = [^xt^,t,t9id ^and ^t = ^^xt^{, 1,Ds91} : 1^], ^{where t}9i.i

is a linear trend starting in ɪ99ɪ:ɪand Ds9F1 is a step dummy starting in ɪ99ɪ:ɪ.

The relations T'∆x_t define the p — s₂ stationary relations between the growth rates,
of which r correspond β'∆x_t + p₀ + p₀₁Ds9F1 and sɪ to β^β₁∆x_t + 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,β_xi) 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'x_t, among the variables
x_t, i.e^ they give a decomposition of the vector x_t into the r + sɪ directions τ = (β, βɪɪ)

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