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

over the sample period and the question is whether this trend should be treated as
stochastic or deterministic. From an economic point of view, a deterministic trend
in relative prices would not be plausible. From a statistical point of view it might,
however, work as a local approximation. We shall include a linear trend in the VAR
and then test whether it can be excluded from the model.

In the present data, the re-unification of Germany is a very significant event which
is likely a priori to have strongly affected the German prices, but not the US. There
are several possibilities, for example: (1) an additive effect on price levels measured
by a step dummy outside the VAR dynamics at the time when the two economies
were merged, (2) an additional innovational effect measured by a step dummy inside
the VAR dynamics, (3) a change in the nominal growth rates which corresponds to a
broken linear trend in the data.⁷

The additive step dummy effect on German prices has been removed prior to the
empirical analysis using a procedure in Nielsen (2004), the remaining two effects will be
properly tested within the model. There are several possibilities for how these effects
will influence the specification of deterministic components:

1. p₁ and p₂ are I(2) with a level shift and linear, but no broken trends, i.e. {p₁ —
b₀D_s91 — b₁t} ~ I(2) and {p₂ — b₂t} ~ I(2), and, provided that p₁ and p₂ share
the same stochastic I(2) trend, {p₁ — p₂ — b₀D_s91 — (b₁ — b₂)t} ~ I(1), i.e. the
price differential adjusted for a linear deterministic trend is I(1). If, in addition,
bi = b2 then {pι — p2 — boDs91} ~ I(1).

2. p₁ and p₂ are I(2) with a level shift and broken linear trends, i.e. {p₁ — b₀D_s91 —
b₁₁t — b₁₂t₉₁.₁} ~ I(2) and {p₂ — b₂₁1} ~ I(2), and, provided that p₁ and p₂ share
the same stochastic I(2) trend, {p₁ — p₂ — b₀D_s91 — (b₁₁ — b₂₁)t — b₁₂t₉₁.₁} ~ I(1),
i.e. the price differential adjusted for a broken linear trend is I(1). If, in addition,
the linear trend is identical in two prices then

^{p1 ^— p2 ^{— b}0^Ds^{91 — b}12^t91:1^} ~ ¹⁽¹⁾.

3. p₁ and p₂ are I(2) with a level shift and linear (or broken linear) trends but the
stochastic I(2) trends do not cancel in p₁ — p₂. In this case, p₁ — p₂ ~ I(2) and
s₁₂,t would also need to be I(2) in order for the ppp_t to be I(1).

We shall allow for a dynamic step dummy and a broken linear trend in the cointe-
gration relations as well as in the data to be able to test the hypothesis whether the
latter is significant (assuming that the step dummy has to be there). The subsequent
results suggest that case 3 works best with our information set, but they also show
that the I(2) trend in pp_t and s₁₂,t are not necessarily identical (which is also what the
graphs showed) implying two stochastic I(2) trends. How can this make sense? The
highly persistent, downward sloping, trend in price differentials (see Figure 1, upper
panel) looks as a near I(2) trend and the long swings in nominal exchange rates can be

⁷For a detailed description of the role of deterministic trends in the I(2) model, see Juselius, 2006,
Chapter 16.

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