Finally, we shall test five hypotheses formulated as a known vector b in τ. If ac-
cepted, they imply the variable in question is at most I(I). If, in addition, the variable
in question is not a vector in β, then it is I(1). None of the variables tested below can
be considered a vector in β, hence, the tests are tests of I(I).
1. ^5 : τ = (bi, bι±φ) where b1 = [1, 0, 0, 0, 0, 0, 0, 0, 0], i.e. we test whether relative
prices is a unit vector in τ which, if accepted, would imply that ppt is I(1). The
test strongly rejected based on y2(4) = 55.56 [0.00] .
2. ^5 : τ = (b2, b^±φ) where b2 = [0,1, 0, 0, 0, 0, 0, 0, 0], i.e. we test whether nominal
exchange rate is a unit vector in τ which, if accepted, would imply that si2,t is
(at most) I(1). The test is rejected based on y2(4) = 9.76 [0.04].
3. ^6 : τ = (b3,b3χ<^>) where b3 = [1, —1, 0, 0, 0, 0, 0, 0, 0], i.e., we test whether the
real exchange rate is a unit vector in τ which, if accepted, would imply that pppt
is I(1). The test is accepted based on y2(4) = 4.90 [0.30] . Thus, real exchange
rates can be approximately considered an I(1) process.
4. H7 : τ = (b3,b3χ<^>) where b4 = [0, 0, 0,1, —1,0, 0, 0, 0], i.e., we test whether
the bond rate spread is a unit vector in τ which, if accepted, would imply that
bi,t — b2,t is I(1). The test is accepted based on y2(4) = 3.65 [0.46] . Thus, bond
rate differential can be approximately considered an I(1) process.
5. H7 : τ = (b3, b3χ<^>) where b5 = [0, 0, 0, 0, 0,1, —1, 0, 0], i.e., we test whether the
short spread is a unit vector in τ which, if accepted, would imply that si,t — s2,t
is I(1). The test is only borderline accepted based on y2(4) = 8.43 [0.08] . Thus,
short spread can be approximately considered an I(1) process, but barely so.
These tests provide an approximate description of the properties of the data and
should not be confused with testing structural hypotheses, which is the topic of the next
section. For example, the result that pppt, bi,t — b2,t, and si,t — s2,t are I(1) seems to be
at odds with the previous assumption that the long swings are I(2). The explanation
for this apparent inconsistency is that the I(2) approximation of the long swings trend
is a borderline case. This is because it consists of a unit root together with a large root
of roughly 0.86. Depending on whether size or power is considered more important,
one can interpret 0.86 as a unit root or argue that it is small enough to be different
from one. But regardless of whether one interprets ppp and the interest rate spreads as
I(1) or I(2), the results are inconsistent with the REH monetary model, which implies
that ppp and interest rate spreads are I(0). By contrast, the monetary model with IKE
implies that these variables are both near I(2). Thus, beyond the inability to formally
reject the unit-root hypothesis, the FG model justifies the I(2) interpretation of the
results.
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