Appendix 2.B
Unit Root Tests
The comparison of developments in government gross debt over the last ten years (see
figures below) reveals that debt growth has been entirely nominal in Belgium, Spain,
Ireland, Italy, Netherlands, Finland and Sweden, and has been real in Germany, Greece,
France, Austria, Portugal, UK, US and Japan. Both when measured in nominal and real
terms, government gross debt appear to behave as integrated series in a majority of
countries. As a matter of fact the usual unit root tests are not able to reject the null of
unit root for both nominal and real debt.13
When debt levels are scaled by GDP a somewhat different picture emerges. After being
on a rising path, the debt-to-GDP ratio attained a maximum in most countries around
the mid-nineties and has been on a descendent path since them. The clearest exceptions
appear to be Germany and France (and Japan). Yet, as shown in Table 2.11, the usual
unit root tests, including that by Kwiatkowski et al. (1992), where the null hypothesis is
mean or trend stationary when the residuals are assumed to be AR(1)14 fail to present
unambiguous statistical evidence against the existence of a unit root in the debt-to-GDP
ratios.
Table 2.11 Unit Root Tests for Gross Debt Series in % of GDP, 1977-2005
|
ημ |
∏τ |
W-S |
ADF |
PP | |
|
(1) |
(2) |
(3) |
(4) |
(5) | |
|
Belgium |
0.60 |
0.37 |
-1.71 |
-2.88 |
-0.86 |
|
Germany |
1.46 |
0.10* |
-2.04 |
-2.83 |
-8.77 |
|
Greece |
1.44 |
0.30 |
-0.63 |
0.25 |
0.64 |
|
Spain |
1.15 |
0.28 |
-1.27 |
-1.44 |
0.74 |
|
France |
1.50 |
0.14* |
-1.46 |
-3.14° |
-8.18 |
|
Italy |
1.28 |
0.32 |
-0.67 |
-1.39 |
-0.93 |
|
Ireland |
0.79 |
0.36 |
-1.33 |
-2.81 |
-2.53 |
|
Netherlands |
0.44* |
0.37 |
-2.86 |
-3.18°° |
-1.74 |
|
Austria |
1.36 |
0.33 |
-0.25 |
-0.64 |
-0.86 |
|
Portugal |
0.62 |
0.29 |
-1.65 |
-3.74°° |
-4.74 |
|
Finland |
1.15 |
0.15* |
-2.02 |
-2.21 |
-5.38 |
|
Denmark |
0.35* |
0.29 |
-1.08 |
-2.35 |
-3.66 |
|
Sweden |
0.46* |
0.19 |
-2.15 |
-2.70 |
-5.91 |
|
UK |
0.75 |
0.13* |
-2.63 |
-3.08 |
-8.44 |
|
US |
0.83 |
0.33 |
-2.18 |
-2.29 |
-2.94 |
|
Japan_________ |
1.32 |
0.31 |
-1.32 |
-3.25 |
-2.24 |
Notes: (1) Test of Kwaikowski et al. (null hypothesis mean-stationary): critical value at 5%: 0.46.
(2) Test of Kwaikowski et al. (null hypothesis trend-stationary): critical value at 5%: 0.15.
(3) Weighted-symmetric (null hypothesis unit root).
(4) Augmented Dickey-Fuller (null hypothesis unit root).
(5) Phillips-Perron (null hypothesis unit root).
* The null of mean/trend-stationary can be accepted at 5%.
°°° The null unit root can be rejected at 1% (°° 5%, ° 10%).
13
14
Results available on request from the authors.
The Barlett window, 1 - s/(l + 1), in the test is calculated for l=1 (then s=1).
50