RDP 8903: The Relationship Between Financial Indicators and Economic Activity: Some Further Evidence Appendix A: Tests for Unit Roots and Time Trends
June 1989
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These tests are typically based on an equation of the form:
The null hypothesis is that there is a unit root in y – i.e. that δ = 0. Tests of this hypothesis frequently involve comparing the computed values from the above equation with appropriate critical values (e.g. the, t-statistic for δ is the test-statistic in the Dickey-Fuller test).
If δ = 0, µ ≠ 0, there is a unit root “with drift”; if β ≠ 0, there is a time trend.
Tables A.1, A.2 and A.3 report results of Dickey-Pantula (DP), Stock-Watson (SW) and Dickey-Fuller (DF) tests for the likely presence of two or less unit roots under a series of nested hypotheses on deterministic trends. (The test statistics were computed using a programme developed by Dr Rob Trevor.) All variables except interest rates are tested in natural log form. The test samples correspond to the data lists attached.
Variable | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Test | Real PFD | Real GDP | Nom PFD | Nom GDP | M1 | M3 | Bank lend. | BM | AFI lend | Credit |
Two unit roots vs. at most one | ||||||||||
T | −2.742** | −1.781 ^{+} | 0.373 | 0.201 | 0.236 | −0.298 | 0.465 | −1.151 | 0.919 | 1.476 |
DP | −6.969** | −3.142^{+} | −3.200^{+} | −2.815 | −3.484* | −3.299^{+} | −3.066 | −3.416* | −3.379^{+} | −2.235 |
SW | −119.120** | −95.409** | −64.411** | −91.491** | −41.129** | −34.800** | −24.820* | −33.588** | −17.073 | −16.143 |
DF | −5.120** | −3.313^{+} | −2.544 | −2.675 | −3.920* | −3.182^{+} | −2.480 | −2.708 | −2.942 | −2.575 |
Constant | 4.701** | 3.169** | 3.365** | 3.330** | 3.718** | 2.788** | 2.655** | 2.762** | 2.684** | 2.343* |
DP | −5.632** | −2.714^{+} | −3.357* | −3.024* | −3.503** | −3.336* | −3.029* | −3.252* | −3.111* | −1.740 |
SW | −117.880** | −93.885** | −58.904** | −87.604** | −40.883** | −34.723** | −24.082** | −33.362** | −16.275* | −12.719 |
DF | −4.323** | −2.905* | −2.785^{+} | −2.907* | −3.942** | −3.212* | −2.457 | −2.454 | −2.784^{+} | −2.074 |
One unit root vs. none | ||||||||||
T^{2} | −2.810** | 0.771 | −1.506 | −1.247 | −1.818^{+} | −2.227* | −1.251 | −2.974** | 2.286* | 2.570* |
SW | −24.096 | −7.276 | −3.408 | −4.680 | −12.182 | −10.634 | −8.592 | −15.137 | −8.224 | −6.818 |
T | 1.475 | 2.386* | 1.876^{+} | 1.842^{+} | 2.377* | 2.290* | 2.893** | 2.358* | 1.822^{+} | 1.152 |
DP | −1.165 | −2.804 | −1.744 | −1.869 | −2.227 | −2.262 | −2.220 | −1.196 | −0.342 | −0.747 |
SW | −4.508 | −6.607 | −5.286 | −6.044 | −9.800 | −8.914 | −8.052 | −7.140 | −5.402 | −2.464 |
DF | −1.602 | −2.785 | −1.822 | −1.846 | −2.365 | −2.852 | −2.628 | −1.870 | −1.462 | −0.612 |
Constant | 2.954** | 2.436* | 0.810 | 0.920 | 0.455 | 1.085 | 0.479 | 1.474 | −0.113 | −0.681 |
DP | −3.773** | −2.080 | 0.268 | −0.050 | 0.081 | −0.241 | 0.462 | −1.149 | 1.262 | 1.335 |
SW | −0.995 | −1.135 | 0.072 | 0.024 | 0.027 | −0.141 | 0.112 | −0.147 | 0.153 | 0.302 |
DF | −2.973* | −2.303 | 0.204 | 0.052 | 0.072 | −0.437 | 0.311 | −1.201 | 0.866 | 1.434 |
^{1} An *(**)(+) indicates rejection of the null hypothesis that the estimated coefficient is zero at the one(five)(ten) per cent level. For “T^{2}”, T, and constant, the null is that the coefficient on the trend or drift term is zero. For the other tests the null is that there are two, or one, unit roots. |
Variable | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Test | Real PFD | Real GDP | Nom PFD | Nom GDP | M1 | M3 | Bank lend. | BM | AFI lend | Credit |
Two unit roots vs. at most one | ||||||||||
T | −2.790** | −2.618** | 0.489 | 0.494 | 0.283 | 0.071 | 0.420 | −1.132 | 0.831 | 1.393 |
DP | −5.307** | −4.091** | −2.656 | −2.720 | −2.744 | −2.366 | −2.325 | −2.804 | −2.428 | −1.997 |
SW | −161.900** | −144.180** | −163.660** | −139.550** | −65.158** | −57.648** | −61.208** | −42.167** | −31.195** | −30.395** |
DF | −7.7631** | −5.790** | −4.071** | −3.744* | −4.370** | −3.342^{+} | −2.838 | −3.584* | −2.962 | −2.955 |
Constant | 4.979** | 4.048** | 4.063** | 3.953** | 5.094** | −3.863** | 2.652** | 2.588** | 2.901** | 2.210* |
DP | −4.294** | −3.275* | −2.904* | −2.923* | −2.755^{+} | −2.401 | −2.303 | −2.375 | −2.297 | −1.513 |
SW | −162.140** | −144.180** | −163.630** | −139.310** | −65.134** | −57.530** | −60.406** | −42.444** | −30.536** | −26.140** |
DF | −6.575** | −4.832** | −4.177** | −3.830** | −4.392** | −3.370* | −2.816^{+} | −3.199* | −2.837^{+} | −2.357 |
One unit root vs. none | ||||||||||
T^{2} | −3.284** | −0.682 | −0.636 | −0.957 | −2.225* | −2.432* | −1.185^{+} | −3.850** | 2.122* | 2.888** |
SW | −135.250** | −110.770** | −35.398* | −52.909** | −15.555 | −13.327 | −8.926 | −27.447^{+} | −8.168 | −7.439 |
T | 2.143* | 1.449 | 1.654^{+} | 1.833^{+} | 2.649** | 2.339* | 2.574* | 2.484* | 1.652^{+} | 0.602 |
DP | −1.522 | −1.975 | −1.803 | −1.944 | −2.805 | −3.159^{+} | −2.974 | −1.158 | −1.546 | −0.566 |
SW | −62.560** | −66.172** | −26.435* | −43.733** | −12.628 | −10.431 | −8.336 | −12.142 | −5.635 | −2.136 |
DF | −1.823 | −1.859 | −1.645 | −1.801 | −1.893 | −2.121 | −2.635 | −1.578 | −1.391 | −0.429 |
Constant | 2.828** | 3.057** | 1.066 | 0.931 | 0.377 | 0.938 | 0.482 | 1.272 | −0.062 | −0.807 |
DP | −3.139* | −2.774^{+} | −0.049 | −0.064 | 0.089 | −0.418 | 0.255 | −1.451 | 0.796 | 1.273 |
SW | −1.469 | −3.229 | 0.087 | 0.049 | 0.044 | −0.044 | 0.096 | −0.161 | 0.133 | 0.314 |
DF | −3.055* | −2.953* | 0.336 | 0.347 | 0.148 | −0.179 | 0.262 | −1.179 | 0.779 | 1.362 |
^{1} An **(*)(+) indicates that the null hypothesis that the estimated coefficient is zero is rejected at the one(five)(ten) per cent level. For “T^{2}”, T, and constant, the null is that the coefficient on the trend or drift term is zero. For the other tests the null is that there are two, or one, unit roots. |
Variable | |||||||||
---|---|---|---|---|---|---|---|---|---|
Test | Real Impts | Real Expts | Nom Impts | Nom Expts | 90 day Bill | TWI | GDP Def. | Price Diff. | M7PFD |
Two unit roots vs. at most one | |||||||||
T | −0.866 | 0.040 | 0.235 | 0.976 | −0.172 | −0.895 | 0.026 | −0.352 | 0.306 |
DP | −6.458** | −4.902** | −5.342** | −4.991** | −3.890* | −3.494* | −1.874 | −1.839 | −2.770 |
SW | −74.813** | −115.750** | −78.767** | −104.060** | −79.569** | −98.801** | −64.220** | −74.390** | −45.300** |
DF | −5.445** | −4.785** | −4.973** | −4.532** | −4.898** | −4.112** | −1.966 | −1.786 | −2.415 |
Constant | 3.301** | 3.657** | 4.178** | 4.165** | 0.621 | −0.728 | 1.723^{+} | 2.072* | 2.062* |
DP | −6.334** | −4.936** | −5.404** | −4.809** | −3.922** | −3.303* | −1.886 | −1.836 | −2.895* |
SW | −74.688** | −115.780** | −78.109** | −103.000** | −79.482** | −95.481** | −60.358** | −74.358** | −44.900** |
DF | −5.374** | −4.819** | −5.062** | −4.427** | −4.945** | −3.892** | −1.969 | −1.785 | −2.545 |
One unit root vs. none | |||||||||
T^{2} | −1.019 | −1.016 | 0.390 | −0.361 | −1.529 | −3.317** | −1.143 | 2.118* | 0.646 |
SW | −24.632+ | −25.561+ | −8.731 | −12.474 | −27.270+ | −17.458 | −5.009 | −6.829 | −7.054 |
T | 3.882** | 1.976 | 1.919^{+} | 2.496* | 3.941** | −2.063* | −3.987** | 2.437* | −2.704** |
DP | −3.169+ | −1.806 | −1.606 | −2.353 | −4.008** | −1.519 | −3.561* | −3.525* | −2.449 |
SW | −21.303^{+} | −20.138+ | −7.166 | −11.390 | −25.596* | −7.040 | −5.570 | −7.868 | −8.256 |
DF | −3.993** | −1.944 | −1.821 | −2.377 | −3.753* | −2.099 | −3.837* | −2.943 | −2.852 |
Constant | 1.660^{+} | 0.560 | 0.893 | 0.280 | 1.969* | 1.539 | 1.054 | 0.414 | 0.162 |
DP | −1.688 | −0.353 | 0.346 | 1.014 | −1.395 | −0.571 | −0.099 | −0.974 | −0.235 |
SW | −1.566 | −0.276 | 0.001 | 0.282 | −7.109 | −1.965 | −0.237 | −1.187 | −0.091 |
DF | −1.516 | −0.360 | −0.019 | 0.711 | −1.533 | −1.087 | −0.347 | −1.006 | −0.086 |
^{1} An **(*)(^{+}) indicates that the null hypothesis that the estimated coefficient is zero is rejected at the one(five)(ten) per cent level. For “T^{2}”, T, and constant, the null is that the coefficient on the trend or drift term is zero. For the other tests the null is that there are two, or one, unit roots. |
The interpretation of the tables is as follows. The first row of figures gives the test statistic for the null hypothesis that there is no time trend in a model which assumes two unit roots. A significant value then leads to the first set of unit root tests immediately below. Here the null is that two unit roots exist. Significant values lead to a rejection of the hypothesis. If the statistic in the ‘T’ row is not significant, the null of no time trend is accepted, and the next test is to see whether the constant, or “drift” term, is significant. If so, the “two unit roots with drift” hypothesis can be tested with the next set of statistics. Significant statistics here mean a rejection of the null hypothesis. If the hypothesis of two unit roots is rejected, the next step is to assess the hypothesis of a single unit root, using an analogous procedure, in the lower half of the table.
The results suggest that all the time series have at least one unit root, indicating the need for first-differencing. The evidence on the trend components of the series is mixed, and generally weak, but several series may have first or second-order time trends.