Katarina Juselius 361Table 8.2 Determination of rank in theI(1) modelrp−r τp−r 4 largest characteristic roots0 3 80.06
[57.9]1.0 1.0 1.0 0.75
1232. 65
[36.6]1.0 1.0 0.99 0.53
216. 72
[18.5]1.0 0.99 0.99 0.52
3 0 0.99 0.99 0.98 0.53
Note:95% quantiles in [ ].Tests of pushing and pulling variables
rp 1 p 2 s 12
No levels feedback 1 7.52
[0.01]
16.17
[0.00]
7.58
[0.01]
2 23.83
[0.00]32.74
[0.00]8.66
[0.01]
Pure adjustment 1 21.40
[0.00]11.26
[0.00]34.27
[0.00]
2 2. 74
[0.10]
1. 31
[0.25]
18.74
[0.00]
Note:p-values in [ ].Juselius, 2006, Ch. 8). Therefore, the choice of rank suggested by the trace test
needs to be checked for its consistency with other information in the model, such
as the characteristic roots.
The trace tests reported in Table 8.2 suggest a borderline acceptance ofr= 1
cointegration relation, and hencep−r=2 common stochastic trends or, alterna-
tively, a strong acceptance ofr=2, and hence,p−r=1 common stochastic trend.
Thus, from a statistical point of view, both choices can be defended. Section 8.8
will argue thatr=2 is the theory consistent choice. To find out which choice
is econometrically preferable, we shall check the consistency ofr=1, 2 with the
characteristic roots in the model and with the mean reversion of the cointegration
relations.
An inspection of the characteristic roots of the model shows that there are three
large roots of magnitude 0.99 in the unrestricted model. These are generally indis-
tinguishable from unit roots, so the model seems to contain three unit roots. The
choice ofr=1 leaves one near unit root and the choice ofr=2 two near unit roots
in the model. Section 8.4 showed that, when one or several large roots remain in
the model for any reasonable choice ofr, it is a sign ofI( 2 )behavior in at least one
of the variables.^4
To check the consistency of the results with theI( 2 )model, it is useful to divide
the total number of stochastic trends intoI( 1 )andI( 2 )trends, i.e.,p−r=s 1 +s 2 ,
wheres 1 denotes the number ofI( 1 )trends (unit root processes), ands 2 the number
ofI( 2 )trends (double unit root processes). Three (near) unit roots in the model
would be consistent with either
{
r=0,p−r= 3}
or{
r=1,s 1 =1,s 2 = 1}
{ , whereas
r=2,s 1 =0,s 2 = 1
}
corresponds to two unit roots. Since the latter is less than the
three near unit roots in the model, the choicer=2 would not be consistent with
the empirical information in the data.