364 The Long Swings Puzzle
Table 8.3 The estimated short-run dynamic adjustment structure in
theI( 1 )model
⎡
⎢⎢
⎢⎣p 1 t
p2,t
s12,t⎤
⎥⎥
⎥⎦︸ ︷︷ ︸
I( 1 )=⎡
⎢⎢
⎢⎢
⎣0. 21
[4.50]0. 12
[2.31]0. 01
[2.06]
0. 10
[2.21]0. 52
[10.23]0.00
[0.38]
0.92
[1.15]
−1.44
[−1.59]−0.01
[−0.18]⎤
⎥⎥
⎥⎥
⎦
︸ ︷︷ ︸
1⎡
⎢⎢
⎢⎣p 1 t− 1
p2,t− 1
s12,t− 1⎤
⎥⎥
⎥⎦︸ ︷︷ ︸
I( 1 )+⎡
⎢⎢
⎢⎢
⎣− 0. 01
[−3.92]
− 0. 02
[−5.92]
− 0. 17
[−3.05]⎤
⎥⎥
⎥⎥
⎦
︸ ︷︷ ︸
α[
β′ 1 xt− 1]
︸ ︷︷ ︸
I( 1 )+⎡
⎢⎢
⎢⎣0. 00
[1.77]0. 02
[4.09]
0.00
[0.16]0. 03
[6.21]
0. 01
[2.57]
0. 22
[2.96]⎤
⎥⎥
⎥⎦︸ ︷︷ ︸
#[
Ds91.1
μ 0]
+⎡
⎣ε1,t
ε2,t
ε3,t⎤
⎦︸ ︷︷ ︸
I( 0 )where:β′ 1 xt= 1. 0 p1,t− 0. 81
[−7.76]p2,t+ 0. 18
[4.39]
s12,t− 0. 0022
[−4.81]t91.1+ 0. 0022
[3.96]
t,and:=⎡
⎣1.00
0.12 1.00
−0.58 −0.07 1.00⎤
⎦The estimatedαcoefficients show that German prices and nominal exchange
rates have been equilibrium correcting to the estimatedβrelation whereas US
prices have been increasing in the equilibrium errors. The overall behavior of
the system is nevertheless stable as the other two variables compensate for the
error-increasing behavior of US prices. The estimated coefficients of 1 show that
lagged inflation rates are quite significant in the price equations, whereas the lagged
depreciation/appreciation rate is only significant in the German price equation. As
already demonstrated in section 8.4, the lagged changes of theI( 2 )variables in 1
are needed to achieve stationarity ofβ′ 1 R1,t.
The estimates ofα⊥1,β⊥ 1 andCin the MA representation of theI( 1 )model are
almost all insignificant and are not reported here. This is because the stochastic
trends in theI( 1 )model are measured by the once cumulated residuals, whereas
the data are generated by second-order stochastic trends, measured by the twice
cumulated residuals. Thus, when data areI( 2 )the MA representation of theI( 1 )
model is completely uninformative.
Based on the above results, it would be hard to argue that the data are not empir-
icallyI( 2 ), and the next step is therefore to address the PPP puzzle in the correct
framework of anI( 2 )model.