Palgrave Handbook of Econometrics: Applied Econometrics

Katarina Juselius 375

Table 8.6 The estimated short-run dynamic adjustment structure in the

I( 2 )model

⎡

⎢⎢

⎢⎢

⎣

^2 p 1 t

^2 p2,t

^2 s12,t

⎤

⎥⎥

⎥⎥

⎦

︸ ︷︷ ︸

I( 0 )

=

⎡

⎢⎢

⎢⎢

⎣

− 0. 01

[−4.98]

− 0. 02

[−8.66]

− 0. 13

[−3.41]

⎤

⎥⎥

⎥⎥

⎦

︸ ︷︷ ︸

α

[

β′ 1 xt− 1 +δ′ 1 xt− 1

]

︸ ︷︷ ︸

I( 0 )

+

⎡

⎢⎢

⎢⎢

⎣

− 0. 51

[−11.97]

− 0. 25

[−11.15]

0. 29

[7.25]

−

[−6.51]

0. 14

1.19

[1.66]

0.06

[0.15]

⎤

⎥⎥

⎥⎥

⎦

︸ ︷︷ ︸

ζ

[

β′ 1 xt− 1

β′⊥1,1xt− 1

]

︸ ︷︷ ︸

I( 0 )

+

⎡

⎣

ε1,t

ε2,t

ε3,t

⎤

⎦,

︸ ︷︷ ︸

I( 0 )

where:

β′ 1 xt+δ′ 1 xt=1.0p1,t− 0. 85

[−7.68]

p2,t+ 0. 19

[15.08]

s12,t− 0. 0025

[−5.99]

t91.1+ 0. 0024

[8.34]

t

+ 2. 61 p1,t+ 5. 21 p2,t+ 9. 31 s12,t− 0. 10 Ds91.1

β′⊥ 1 xt− 1 = 1. 01 p1,t+ 1. 0 p2,t− 0. 84 s12,t+ 0. 01 t91.1− 0. 01 t

Note:t-values in [ ].

xt∼I( 2 ), long-run price homogeneity is defined onτ′xt, whereτ′=[β,β⊥ 1 ].
Hence (see Johansen, 2006b), long-run homogeneity onβis a necessary but not
sufficient condition. When tested, long-run price homogeneity ofτ′xtwas strongly

rejected based onχ^2 ( 2 )=22.95[0.00]andβ′⊥ 1 xtcannot be considered homo-
geneous in prices. As Table 8.6 demonstrates, the coefficients to prices onβ⊥ 1 are
proportional to (1, 1) rather than (1,−1). This, of course, is just another piece of
evidence associated with the PPP puzzle.
Table 8.6 also reports the estimates of short-run adjustment dynamics towards the
estimated long-run equilibrium relations. TheI( 2 )model is parameterized accord-
ing to (8.31). We note that theI( 2 )model allows the VAR variables to adjust to
a medium-run equilibrium error,β′⊥ 1
x ̃t− 1 , to a change in the long-run “static
equilibrium” error,β′x ̃t− 1 , and to the long-run “dynamic equilibrium” error,

β′x ̃t− 1 +δ
x ̃t− 1. In this sense, theI( 2 )model offers a much richer dynamic
adjustment structure than theI( 1 )model.
When discussing the adjustment dynamics with respect to the polynomially
cointegrating relations, it is useful to interpret the adjustment coefficientsαandδ
as two levels of equilibrium correction. Consider, for example, the following model
for the variablexi,t:

^2 xi,t=···

∑r

j= 1

αij(δ′jxt− 1 +β′jxt− 1 )+··· (8.35)