RDP 2000-02: Forecasting Australian Economic Activity Using Leading Indicators Appendix B: Unit Root and Cointegration Tests
April 2000
- Download the Paper 339KB
Quarterly series
Unit root tests for the WM index and GDP, both in log levels, are reported below for lag lengths four to ten. In both cases, the estimated regression contains a constant and a trend. Two tests are presented; a standard augmented Dickey-Fuller test statistic, the t-test in the table, and a joint test against a trend stationary alternative, the F-test in the table. These are described in Hamilton (1994, pp 528–529). We cannot reject the null hypothesis of a unit root in the WM index or GDP. In the case of GDP, however, this hypothesis only holds up weakly against a trend stationary alternative (for some lag lengths, we can reject the F-test statistic between the five and ten per cent level).
p=4 | p=5 | p=6 | p=7 | p=8 | p=9 | p=10 | |
---|---|---|---|---|---|---|---|
WM | |||||||
t-test | −3.3873 | −2.9226 | −2.6897 | −2.5220 | −2.4522 | −2.2213 | −2.2570 |
F-test | 5.7436 | 4.3011 | 3.6963 | 3.3924 | 3.2969 | 2.6891 | 2.6869 |
GDP | |||||||
t-test | −2.0007 | −2.8045 | −2.7223 | −2.8465 | −2.6778 | −2.9985 | −2.8840 |
F-test | 2.9186 | 5.1467 | 5.4152 | 5.9688 | 5.3571 | 5.8254 | 5.0390 |
Notes: The 5 per cent significance level for the t-test is −3.45 (Hamilton 1994, Table B6, Case 4, p 763). The 5 and 10 per cent significance levels for the F-test are 6.49 and 5.47 respectively (Hamilton 1994, Table B7, Case 4, p 764). In both cases the null hypothesis is a unit root. All test regressions include a constant and a trend. The t-test is the augmented Dickey-Fuller test statistic; The F-test is a joint test against a trend stationary alternative. Both are described in Hamilton (1994, pp 528–529). |
Under the assumption that the WM index and GDP are both I(1), we present Johansen's tests for cointegration between these two variables. These results are reported in the Table below. We find we cannot reject the hypothesis that there is a cointegrating vector between the WM index and GDP.
Cointegrating vector: [ln(WM) γln(GDP)] | ||||||||
---|---|---|---|---|---|---|---|---|
p=4 | p=5 | p=6 | p=7 | p=8 | p=9 | p=10 | ||
γ | −0.8547 | −0.8584 | −0.8619 | −0.8684 | −0.8780 | −0.8859 | −0.8926 | |
Likelihood ratio test 1 | ||||||||
(Trace statistic) | ||||||||
H_{0}: r≤1 (g=1) | {3.962} | 1.0870 | 0.6715 | 0.5502 | 0.2699 | 0.1174 | 0.0216 | 0.0806 |
H_{1}: r=2 (g=0) | ||||||||
H_{0}: r=0 (g=2) | {15.197} | 17.4739 | 17.7023 | 20.1150 | 20.5906 | 20.2534 | 18.3504 | 14.8846 |
H_{1}: r=2 (g=0) | ||||||||
Likelihood ratio test 2 | ||||||||
(Largest Eigenvalue) | ||||||||
H_{0}: r=0 (g=2) | {14.036} | 16.3869 | 17.0307 | 19.5648 | 20.3208 | 20.1360 | 18.3288 | 14.8040 |
H_{1}: r=1 (g=1) | ||||||||
H_{0}: r≤1 (g=1) | {3.962} | 1.0870 | 0.6715 | 0.5502 | 0.2699 | 0.1174 | 0.0216 | 0.0806 |
H_{1}: r=2 (g=0) | ||||||||
Notes: 5 per cent critical values are reported in braces and are taken from Hamilton (1994, Tables B10 (LR1) and B11 (LR2), Case 3, pp 767–768). For both test statistics, r is the number of cointegrating vectors and g is the number of random walks (or stochastic trends). All auxiliary regressions include a constant. |
Monthly series
Unit root tests for the WM index and employment in log levels and unemployment in levels are presented in the table below for lag lengths 12 to 16. The tests are as described for the quarterly series. We test for a unit root in the WM index over each of the sample periods for which it is used. There is evidence of sixth order serial correlation in the test regressions for employment with lag lengths less than 13 and in the regressions for unemployment with lag lengths less than 14. There is evidence of sixth order serial correlation in the test regressions for the WM index for the sample starting in 1959 with lag lengths less than 13 and for the sample starting in 1966 with lag lengths less than 14. We cannot reject the null hypothesis of a unit root in employment, unemployment or in the WM index over the sample 1966 to 1999. This is also the case for the WM index over the longer sample, although the evidence is weak at shorter lag lengths.
p=12 | p=13 | p=14 | p=15 | p=16 | |
---|---|---|---|---|---|
Sample: 1959:M9–1999:M4 | |||||
WM | |||||
t-test | −3.3013 | −3.4897 | −2.8389 | −2.8592 | −2.7331 |
F-test | 5.6508 | 6.2665 | 4.2139 | 4.2079 | 3.8182 |
Unemployment | |||||
t-test | −2.6112 | −2.1647 | −2.5141 | −2.4907 | −2.6459 |
F-test | 3.5876 | 2.5286 | 2.4866 | 3.2413 | 3.5982 |
Sample: 1966:M7–1999:M4 | |||||
WM | |||||
t-test | −3.0354 | −3.2405 | −2.6415 | −2.6868 | −2.5782 |
F-test | 4.7038 | 5.3391 | 3.6443 | 3.8175 | 3.5142 |
Employment | |||||
t-test | −3.3872 | −2.8098 | −2.6774 | −3.0769 | −3.1252 |
F-test | 5.7930 | 4.0158 | 3.6501 | 4.7852 | 4.9388 |
Notes: The 5 per cent significance level for the t-test is −3.45 (Hamilton 1994, Table B6, Case 4, p763). The 5 and 10 per cent significance levels for the F-test are 6.49 and 5.47 respectively (Hamilton 1994, Table B7, Case 4, p 764). In both cases the null hypothesis is a unit root. All test regressions include a constant and a trend. The t-test is the augmented Dickey-Fuller test statistic; The F-test is a joint test against a trend stationary alternative. Both are described in Hamilton (1994, pp 528–529). |
Since the WM index, employment and unemployment are I(1), we test for cointegration. In the following tables, we test for cointegration between the WM index and unemployment and the WM index and employment. In both cases, we cannot reject the null hypothesis of no cointegration.
Cointegrating vector: [ln(WM) γunemp] | ||||||
---|---|---|---|---|---|---|
p=12 | p=13 | p=14 | p=15 | p=16 | ||
γ | −0.1178 | −0.1174 | −0.1152 | −0.1164 | −0.1150 | |
Likelihood ratio test 1 | ||||||
(Trace statistic) | ||||||
H_{0}: r≤1 (g=1) | {3.962} | 0.2209 | 0.3661 | 0.2964 | 0.1372 | 0.0349 |
H_{1}: r=2 (g=0) | ||||||
H_{0}: r=0 (g=2) | {15.197} | 6.8828 | 5.8542 | 7.2608 | 5.7776 | 5.7289 |
H_{1}: r=2 (g=0) | ||||||
Likelihood ratio test 2 | ||||||
(Largest Eigenvalue) | ||||||
H_{0}: r=0 (g=2) | {14.036} | 6.6619 | 5.4881 | 6.9644 | 5.6404 | 5.6940 |
H_{1}: r=1 (g=1) | ||||||
H_{0}: r≤1 (g=1) | {3.962} | 0.2209 | 0.3661 | 0.2964 | 0.1372 | 0.0349 |
H_{1}: r=2 (g=0) | ||||||
Notes: 5 per cent critical values are reported in braces and are taken from Hamilton (1994, Tables B10 (LR1) and B11 (LR2), Case 3, pp 767–768). For both test statistics, r is the number of cointegrating vectors and g is the number of random walks (or stochastic trends). All auxiliary regressions include a constant. |
Cointegrating vector: [ln(WM) γln(empl)] | ||||||
---|---|---|---|---|---|---|
p=12 | p=13 | p=14 | p=15 | p=16 | ||
γ | −1.6190 | −1.6090 | −1.6145 | −1.6295 | −1.6273 | |
Likelihood ratio test 1 | ||||||
(Trace Statistic) | ||||||
H_{0}: r≤1 (g=1) | {3.962} | 0.0110 | 0.0563 | 0.0005 | 0.0139 | 0.0039 |
H_{1}: r=2 (g=0) | ||||||
H_{0}: r=0 (g=2) | {15.197} | 11.1306 | 12.8262 | 12.1212 | 9.4628 | 10.1649 |
H_{1}: r=2 (g=0) | ||||||
Likelihood ratio test 2 | ||||||
(Largest Eigenvalue) | ||||||
H_{0}: r=0 (g=2) | {14.036} | 11.1196 | 12.7699 | 12.1208 | 9.4489 | 10.1609 |
H_{1}: r=1 (g=1) | ||||||
H_{0}: r≤1 (g=1) | {3.962} | 0.0110 | 0.0563 | 0.0005 | 0.0139 | 0.0039 |
H_{1}: r=2 (g=0) | ||||||
Notes: 5 per cent critical values are reported in braces and are taken from Hamilton (1994, Tables B10 (LR1) and B11 (LR2), Case 3, pp 767–768). For both test statistics, r is the number of cointegrating vectors and g is the number of random walks (or stochastic trends). All auxiliary regressions include a constant. |