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

20 г-                            Polynomial cointegration relation 1

1980                1985                1990                1995

1980                1985                1990                1995

3∙pdf                     1980              1985               1990              1995

Figure 2: The graphs of the three polynomial cointegration relations. Upper panel
describes the IKE relation, the middel panel the inflation expectations relation, and
the lower panel the German inflation rate relation.

β'₂xt + δ'₂∆xt = {(½ - s₂) + 0.60∆p₂ - O.51∆pι - O.17∆sι₂ + 0.00sι₂ + 0.00⅛.1} (23)

The third relation, essentially a relation for German inflation rate, is similar to the
relation found in Juselius and MacDonald (2006) and describes the latter as (almost)
homogeneously related to US inflation rate, German short-term interest rate, and the
change in the Dmk/$ rate:

β3x_t + J₃∆x_t = {1.31∆p₁ - 0.31∆p₂ - 0.74s₁ - 0.07∆s₁₂ + 0.00pp - 0.00t₉₁.₁} (24)

All three relations contain a tiny, but significant, trend effect which is more difficult
to interpret. The most likely explanation is the usual one that the linear trend effect in
the relations is a proxy for some information not included in the analysis. For example,
the small trend effect in (22) might account for some perceived productivity differential
between the two economies. In (23) the re-unification trend might be a proxy for a
change in the market’s re-assessment of the riskiness of the nominal Dmk/$ rate. In
(24) the trend together with the pp may imply that German inflation rate, in addition
to following the US inflation rate, the short-term interest rate, and the change in the
Dmk/$ rate, has exhibited a long-run adjustment to trend-adjusted relative prices.
Figure 2 shows that the three polynomially cointegrating relations are very stationary.
!

The β_±11x_t relations are CI(2,1) cointegrating relations which only become sta-

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tionary by differencing. Thus, β_x1∆x_t could be interpretable as partially specified

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