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 + &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)

δx-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



More intriguing information

1. The Economic Value of Basin Protection to Improve the Quality and Reliability of Potable Water Supply: Some Evidence from Ecuador
2. Education and Development: The Issues and the Evidence
3. Population ageing, taxation, pensions and health costs, CHERE Working Paper 2007/10
4. Manufacturing Earnings and Cycles: New Evidence
5. The name is absent
6. Private tutoring at transition points in the English education system: its nature, extent and purpose
7. TECHNOLOGY AND REGIONAL DEVELOPMENT: THE CASE OF PATENTS AND FIRM LOCATION IN THE SPANISH MEDICAL INSTRUMENTS INDUSTRY.
8. ESTIMATION OF EFFICIENT REGRESSION MODELS FOR APPLIED AGRICULTURAL ECONOMICS RESEARCH
9. The name is absent
10. THE AUTONOMOUS SYSTEMS LABORATORY