mid seventies until the beginning of the EMU. Section 10 reports the estimates of
the long-run common trends and discusses how they have pushed the variables of this
system. Section 11 concludes.
2 Theoretical background for I(2) analysis
For simplicity, the discussion of the various components in the I(2) model will be based
on the VAR(3) model formulated in acceleration rates, changes and levels:
Δ2Xt = Γ∣∆'<v I ■ ΓΔ.- I +∏¾-ι (1)
+ Φs^s,t + &pDPjt + Φ∕.,∙ Dtrjι + F0 + F1t + F2t91:1 + ɛi,
where x't = ∖ppt,S12,t, Δp2,t,b1jt,b2,t, Sιjt, S2,t], with ppt = (pi - p2)t describing the log
of relative prices, S12jt the Dmk/$ rate, b1jt,b2jt the long-term bond rates, s1jt,s2jt the
short-term interest rates, DSjt is a step dummy (...0,0,1,1,...), DPjt is a permanent
impulse dummy (...0,0,1,0,0...), Dtrjt is a transitory impulse dummy (...0,0,1,-1,0,0...),
and all parameters are unrestricted.
Similar to the I(1) model, we define the concentrated I(2) model:
Ro,t = γ R1,t + ∏R∙2!,t + ɛt (2)
where R0jt, R1jt, and R2jt are defined by:
δ xt = B1δ χt-1 + B2DSjt + B3DPjt + B1 Dtrjt + ^j^ (3)
δx⅛-1 = b5δ xt-1 + BSDSjt + B7DPjt + B>&Dtrjt + R1jt, (4)
and
xt,-1 = b9δ xt-1 + B10DSjt + B11DPjt + B12Dtrjt + R2jt- (3)
where xt = ∖x't, t, t91:1] indicates that Xt has been augmented with a trend, and a broken
trend t91:1. Note that we need to define three types of ’residuals’ in the I(2) model
rather than two in the I(1) model. Similar to the I(1) model all estimation and test
procedures are based on (2).
The hypothesis that xt is I(1) is formulated as a reduced rank hypothesis
∏ = aβ' , where a, β are p × r (6)
implicitly assuming that Γ is unrestricted. The hypothesis that xt is I(2) is formulated
as an additional reduced rank hypothesis
α,hΓ^ɪ = ξη', where ξ, η are (p — r) × S1. (7)
Thus, the Γ matrix is no longer unrestricted in the I(2) model. The first reduced rank
condition (6) is associated with the variables in levels and the second (7) with the
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