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

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 + &p^DPjt + Φ∕.,∙ Dtr_jι + F₀ + F₁^t + F₂^t91:1 + ɛi^,

where x'_t = ∖ppt,S12,t, Δp2,t,b1_jt,b2,t, Sι_jt, S2,t], with pp_t = (pi - p₂)_t describing the log
of relative prices, S_12jt the Dmk/$ rate, b_1jt,b_2jt the long-term bond rates, s_1jt,s_2jt the
short-term interest rates, D_Sjt is a step dummy (...0,0,1,1,...), D_Pjt is a permanent
impulse dummy (...0,0,1,0,0...), D_trjt 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 R_0jt, R_1jt, and R_2jt are defined by:

^{δ x}t = B1^{δ χ}t-1 ^{+ B}2^DSjt ⁺ B₃^D_Pjt ⁺ B₁ ^Dtrjt ⁺ ^j^             ⁽³⁾

^δx⅛-1 = ^b5^{δ x}t-1 ^{+ B}S^DSjt ^{+ B}7^DPjt ^{+ B}>&^Dtrjt ^{+ R}1jt^{,             (4)}

and

^xt,-1 = ^b9^{δ x}t-1 ^{+ B}10^DSjt ^{+ B}11^DPjt ^{+ B}12^Dtrjt ^{+ R}2jt-             ⁽³⁾

where x_t = ∖x'_t, t, t_91:1] indicates that X_t has been augmented with a trend, and a broken
trend t_91:1. 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 x_t 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 x_t 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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