4.4 Tests on τ
We consider first the same restriction on all vectors in τ, that is,
τ = Hφ
(20)
where H is p × m is known and φ is an m × (r + si) matrix of unknown parameters.
An equivalent formulation is R'τ = 0, where R = H±.
The other hypothesis corresponds to (15), that is,
τ = (b,brψ)
(21)
where b is p × 1 and known and ψ is a (p — 1) × (r + si — 1) matrix of unknown
parameters.
The test statistic for the first test is asymptotically distributed with degrees of
freedom (p—m)(r+si), and, in general, the test for the second one is also asymptotically
distributed as χ2 with s2 — 1 degrees of freedom. There is, however, one case when the
asymptotic distribution is not a χ2 distribution. This is when the vector b is a vector
in β, that is, when the hypothesis β = (b, b±ξ) is satisfied.
This problem can be avoided by first testing the hypothesis β = (b, b±ξ) and, if
accepted, then we have that b is a vector in τ. If it is rejected, we can test τ = (b, b±ψ)
and apply the χ2 distribution because we have checked that b ^ sp(β).
The above problem is related to the conditions (36) and (37) in the Appendix
which have to be checked for this case. It is shown in Johansen (2006) that for a vector
b, inference on the hypothesis τ = (b, bχψ) is χ2 if the dimension of sp[β)∩sp[b) is
max(0,1 — Si). Thus if si ≥ 1 then the condition states that b 0sp(β).
5 An ocular inspection of the persistent behavior
in the data
An ocular inspection of the data offers a first impression of the time series properties
of the nominal variables and illustrates their tendency to undergo long swings. Figure
1, upper panel, shows the graphs of the price differential and nominal exchange rate.
There are three features in the upper panel that are important to notice: (1) the
downward sloping trend in price differentials which should be considered a stochastic
trend (as a deterministic trend would not make sense); (2) the big swings in the nominal
exchange rate that evolve around a downward sloping trend which looks very similar
to the one behind relative goods prices; and (3) a possible change in the slope of
the time trend of goods prices around 1991 (together with a shift in the level) and
possibly one around 1980-81. Figure 1, lower panel, shows the mean and range adjusted
ppp = pi — p2 — si2 and the real bond rate differential, (bi — ∆pi) — (b2 — ∆p2), where
the latter is given as a 12 months moving average. The close co-movements between
the two series are quite remarkable and have previously been found among others
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