RDP 9509: Australian Wage and Price Inflation: 1971–1994 Appendix b: Integration Tests of the Data
November 1995
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The following tables present tests of the time series characteristics of the data. Table B1 shows the standard augmented Dickey Fuller (Said and Dickey (1984)) (ADF) test where a unit root null hypothesis is tested against a stationary alternative. These result are compared with Kwiatkowski, Phillips, Schmidt and Shin (1992) (KPSS) tests in Table B2 for which the null hypothesis of stationarity is tested against the unit root alternative.
Variable> | Φ_{1} | Φ_{2} | Φ_{3} | |||
---|---|---|---|---|---|---|
prices^{fc} | 5.56# | 3.71 | 5.31 | 0.05 | −3.26^{*} | −0.28 |
Δ prices^{fc} | 10.37^{**} | 8.50^{**} | 12.69^{**} | −2.39* | −4.54^{**} | −4.81^{**} |
prices | 5.11 | 3.71 | 4.85 | 0.02 | −3.13^{*} | −0.73 |
Δ prices | 2.27 | 12.09^{**} | 18.11^{**} | −0.99 | −2.12 | −5.87^{**} |
wages | 111.73^{**} | 74.45^{**} | 14.79^{**} | 1.16 | −5.39^{**} | −0.27 |
Δ wages | 19.12^{**} | 19.88^{**} | 29.81^{**} | −3.49^{**} | −6.18^{**} | −7.72^{**} |
unit labour costs | 49.04^{**} | 32.79^{**} | 9.12^{**} | 0.98 | −4.20^{**} | −0.13 |
Δ unit labour costs | 25.55^{**} | 23.46^{**} | 35.17^{**} | −4.96^{**} | −7.15^{**} | −8.38^{**} |
import prices | 22.14^{**} | 14.73^{**} | 3.56 | 3.20 | −2.63# | −0.15 |
Δ import prices | 24.60^{**} | 18.33^{**} | 27.49^{**} | −5.65^{**} | −7.01^{**} | −7.41^{**} |
petrol prices | 8.79^{**} | 5.84^{*} | 1.38 | 3.45 | −1.63 | −0.75 |
Δ petrol prices | 60.49^{**} | 42.29^{**} | 63.43^{**} | −9.46^{**} | −11.00^{**} | −11.26^{**} |
output gap | 7.48^{**} | 5.37^{*} | 8.05^{*} | −0.01 | −3.87^{**} | −3.97^{*} |
Δ output gap | 51.36^{**} | 33.88^{**} | 50.79^{**} | −10.19^{**} | −10.13^{**} | −10.08^{**} |
inside unemployment | 7.54^{**} | 4.99^{*} | 7.48^{*} | −0.81 | −3.88^{**} | −3.87^{*} |
Δ inside unemployment | 21.90^{**} | 14.68^{**} | 21.92^{**} | −6.66^{**} | −6.60^{**} | −6.61^{**} |
strikes | 30.59^{**} | 20.42^{**} | 30.63^{**} | −7.86^{**} | −7.82^{**} | −7.82^{**} |
Δ strikes | 16.98^{**} | 11.37^{**} | 17.05^{**} | −5.87^{**} | −5.83^{**} | −5.84^{**} |
rer | 1.95 | 1.55 | 2.33 | −0.05 | −1.97 | −2.15 |
Δ rer | 42.95^{**} | 28.36^{**} | 42.54^{**} | −9.32^{**} | −9.27^{**} | −9.22^{**} |
income share | 2.04 | 4.38# | 6.57# | −0.21 | −2.02 | −3.58^{*} |
Δ income share | 15.48^{**} | 10.39^{**} | 15.43^{**} | −5.60^{**} | −5.56^{**} | −5.55^{**} |
Notes: The likelihood ratio tests are: |
Constant | Constant and trend | |||||
---|---|---|---|---|---|---|
Lag | Lag | |||||
Variable | 0 | 4 | 8 | 0 | 4 | 8 |
prices^{fc} | 9.33^{**} | 1.96^{**} | 1.13^{**} | 2.07^{**} | 0.45^{**} | 0.28^{**} |
Δ prices^{fc} | 3.67^{**} | 1.15^{**} | 0.74^{**} | 0.29^{**} | 0.13# | 0.10 |
prices | 9.33^{**} | 1.96^{**} | 1.13^{**} | 2.09^{**} | 0.46^{**} | 0.28^{**} |
Δ prices | 4.22^{**} | 1.15^{**} | 0.73^{**} | 0.37^{**} | 0.13# | 0.10 |
wages | 9.16^{**} | 1.93^{**} | 1.13^{**} | 2.11^{**} | 0.46^{**} | 0.28^{**} |
Δ wages | 2.46^{**} | 1.04^{**} | 0.71^{*} | 0.07 | 0.05 | 0.04 |
unit labour costs | 9.07^{**} | 1.91^{**} | 1.11^{**} | 1.93^{**} | 0.43^{**} | 0.26^{**} |
Δ unit labour costs | 1.64^{**} | 0.79^{**} | 0.60^{*} | 0.06 | 0.04 | 0.04 |
import prices | 9.06^{**} | 1.90^{**} | 1.11^{**} | 1.88^{**} | 0.41^{**} | 0.25^{**} |
Δ import prices | 0.83^{**} | 0.45# | 0.40# | 0.09 | 0.06 | 0.07 |
petrol prices | 9.14^{**} | 1.91^{**} | 1.10^{**} | 1.96^{**} | 0.43^{**} | 0.26^{**} |
Δ petrol prices | 0.33 | 0.40# | 0.35# | 0.07 | 0.10 | 0.11 |
output gap | 1.45 | 0.38# | 0.29 | 0.33^{**} | 0.09 | 0.07 |
Δ output gap | 0.04 | 0.04 | 0.05 | 0.04 | 0.04 | 0.05 |
inside unemployment | 0.25 | 0.07 | 0.06 | 0.24^{**} | 0.07 | 0.06 |
Δ inside unemployment | 0.05 | 0.03 | 0.05 | 0.05 | 0.03 | 0.04 |
strikes | 0.10 | 0.08 | 0.08 | 0.05 | 0.04 | 0.05 |
Δ strikes | 0.01 | 0.04 | 0.06 | 0.01 | 0.03 | 0.05 |
rer | 1.51^{**} | 0.35# | 0.23 | 0.44^{**} | 0.11 | 0.07 |
Δ rer | 0.07 | 0.06 | 0.05 | 0.07 | 0.06 | 0.05 |
income share | 2.06^{**} | 0.51^{*} | 0.36# | 0.61^{**} | 0.16^{*} | 0.12# |
Δ income share | 0.07 | 0.05 | 0.06 | 0.05 | 0.03 | 0.04 |
Notes: For this test the series, y_{t}, is expressed
as y_{t} = ξt + r_{t} +
ε_{t}, where r_{t} =
r_{t−1} + u_{t} and . Then the null hypothesis of stationarity is the
test that . The critical values, with the inclusion of a
constant at the 1%, 5%, and 10% levels of significance, are 0.739, 0.463 and 0.347
respectively. The critical values, with the inclusion of a constant and trend at the
1%, 5%, and 10% levels of significance, are 0.216, 0.146 and 0.119
respectively. |
Using the augmented Dickey Fuller tests and a 5% level of significance, the level of market prices, prices at factor costs, import prices and petrol prices all accept the null hypothesis of non-stationarity. The first difference of these series reject the null hypothesis in favour of the stationary alternative. The level of the output gap, inside unemployment, and strike terms are all, as expected, found to be stationary. For each of these series, the results of the KPSS tests for a lag length of 8 accord with the above conclusions using a 5% level of significance (see notes for Table B2).
There are, however, four contentious results. While the ADF tests suggest wages and unit labour costs are stationary around a drift term, the null hypothesis of stationarity is rejected using the KPSS tests. Both these series are treated as non-stationary in the estimation. Of particular interest is the characterisation of the real exchange rate and labour's income share. In the paper both these series are characterised as stationary. However, the statistical tests produce conflicting results. In the ADF tests, both series do not reject the unit root hypothesis. The stationary null hypothesis in the KPSS tests is also not rejected at the 5% significance level. While the statistical results are ambiguous, the conclusion of stationarity follows from the graphical analysis and can be supported theoretically.