372 The Long Swings Puzzle
accordingly. Juselius (2006, Ch. 17) proposed the following interpretation:
1.β′xt+δ′xtasr dynamic long-run equilibrium relations, or alternatively when
r>s 2
- β′ 0 xtasr−s 2 static long-run equilibrium relations, and
- β′ 1 xt+δ 1 xtass 2 dynamic long-run equilibrium relations,
2.τ′xtasmedium-run equilibrium relations.
8.8 Two hypothetical scenarios
To be able to structure and interpret the empirical VAR results, it is useful to formu-
late a scenario for what we would expect to find in the VAR model, provided the
reality is in accordance with the assumptions of the theoretical model. For exam-
ple, the first scenario below is specified for the hypothesis:{pppt∼I( 0 ), prices are
pushing and the nominal exchange rate is pulling}under the assumption thatxt
is empiricallyI( 2 ).
We shall discuss the following two cases: (1)r=2, which corresponds to the
theory consistent case, and (2)r=1, which is what we find in the data. In both
cases it will be assumed that long-run price homogeneity holds, i.e.,β′⊥ 2 =[c,c,0].
Case 1 {r=2,s 1 =0,s 2 = 1 }is consistent with:
⎡
⎢
⎣
p1,t
p2,t
s12,t⎤
⎥
⎦=⎡
⎢
⎣c
c
0⎤
⎥
⎦∑tj= 1∑ji= 1u1,i+⎡
⎢
⎣b 1
b 2
b 3⎤
⎥
⎦∑ji= 1u1,i+⎡
⎢
⎣ε1,t
ε2,t
ε3,t⎤
⎥
⎦. (8.34)It is easy to see that (p1,t−p2,t)∼I( 1 )and (p1,t−p2,t−s12,t)∼I( 0 )if (b 1 −
b 2 )=b 3. When the nominal exchange rate is adjusting (and price shocks are
pushing), one would have it thatu1,t=α′⊥ 1 εtwithα′⊥ 1 =[a 1 ,a 2 ,0]. This scenario
would imply two cointegrating relations, one of which is directly cointegrating,
becauser−s 2 =1, and the other of which is polynomially cointegrating, because
s 2 =1. It is easy to show that the directly cointegrating relation is theppprelation,
i.e., (p1,t−p2,t−s12,t)∼I( 0 ). The polynomially cointegrated relation is more
difficult to see and it is helpful to examine the system based on the nominal-to-real
transformation (Kongsted, 2005):^9
⎡
⎢
⎣
p1,t−p2,t
p1,t
s12,t⎤
⎥
⎦=⎡
⎢
⎣b 1 −b 2
c
b 3⎤
⎥
⎦∑ji= 1u1,i+⎡
⎢
⎣̃ε1,t
̃ε2,t
ε3,t⎤
⎥
⎦.It is now straightforward to show that
{
p1,t−p2,t+ωp1,t}
∼I( 0 ),ifc=−(b 1 −b 2 /ω). Alternatively, ifc=−b 3 /ω, then
{
s12,t+ωp1,t}
∼I( 0 ). In both cases the
polynomially cointegrating relation can be thought of as a dynamic equilibrium
relation describing how the inflation rate adjusts when relative prices have been
pushed apart, i.e.,p1,t=− 1 /ω(p1,t−p2,t). It simply states the obvious, that the