Katarina Juselius 367to the other is difficult to know, but we need to pay sufficient attention to this
question, as the choice is likely to influence the empirical results significantly.
In the present data, the reunification of Germany is likely to have affected
German prices significantly, but not US prices. The raw data exhibit an extra-
ordinary large shock in^2 p1,tdue to the reunification in 1991:1. A big impulse in
^2 p1,tcumulates to a level shift inp1,t, and double cumulates to a broken linear
trend inp1,t. Thus, accounting for the extraordinary large shock at 1991:1 with a
blip dummy in^2 p1,t, a shift dummy inp1,tis econometrically consistent with
broken linear trends in prices. Because such a broken linear trend may or may
not cancel inβ′xt, the model should be specified to allow for a (testable) broken
linear trend inβ′xt. Likewise, the level shift may (or may not) cancel inδ′xtor
τ′xt. Thus the model specification should allow for this possibility. Inspecting
the graphs in Figure 8.1 shows an increasing trend in both prices and a downward
sloping trend in relative prices, and the question is whether the latter is canceled
by cointegration with the nominal exchange rate.
Whatever the case, quadratic or cubic trends will be excluded from the outset
and the model specification should account for this.
To understand the role of the deterministic terms in theI( 2 )model, it is useful
to specify the mean of the stationary parts of (8.16) allowing for the above effects
(so that they can be tested), while at the same time excluding cubic or quadratic
trend effects.
The mean of^2 xtshould be allowed to contain the impulse dummies as these
do not double cumulate to quadratic trends, i.e.:
E^2 xt=pDp,t.The mean of the polynomially cointegrated relations should be allowed to have
a trend and a broken linear trend inβ′xtand a constant and a shift dummy in
δ′xt, i.e.:
E(β′xt+δ′xt)=ρ 0 t+ρ 01 t91.1+γ 0 +γ 01 Ds91.1t. (8.25)
The mean of the difference stationary relationsτ′xtshould be allowed to
contain a shift dummy and a constant, i.e.:
E(τ′xt)=ω 0 +ω 01 Ds91.1t.The question is now how to restrictμ 0 ,μ 01 ,μ 1 , andμ 11 in (8.16)^7 to allow for
the deterministic components in the above mean values while suppressing any
quadratic or cubic trend effects in the model. The general idea will only be demon-
strated for the constant termμ 0 and the linear termμ 1 , as the procedure is easily
generalized to the step dummy and the broken trend. A more detailed discussion
is given in Juselius (2006, Ch. 17).
First, the constant termμ 0 is decomposed into three components proportional
toα,α⊥ 1 andα⊥ 2 :
μ 0 =αγ 0 +α⊥ 1 γ 1 +α⊥ 2 γ 2. (8.26)