RDP 9610: Share Prices and Investment Appendix A: Unit Root Tests
December 1996
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Since unit root tests are widely recognised as having low power, we use two different test procedures – the Augmented Dickey-Fuller (ADF) test described in Said and Dickey (1984) and the Phillips and Perron (1988) Zt test[24]. Both tests are based on the testing strategy recommended by Perron (1988).
The tests are conducted over the estimation period 1960:Q1–1996:Q1. The tests share the same MacKinnon (1991) critical values which, with the inclusion of a constant at the 1 per cent and 5 per cent levels of significance, are 3.48 and 2.88, respectively. The critical values with the inclusion of a constant and trend at the 1 per cent and 5 per cent significance levels are 4.03 and 3.44, respectively. In both cases, the null hypothesis is non-stationarity. Apart from the 10-year bond yield, all the variables are in logs.
The results for the level of each series are presented in Table A1. The level of the real return on capital, output gap and real equity raisings are determined to be stationary. All the remaining variables accept the null hypothesis of non-stationarity. Subsequent tests confirmed that none of the series are I(2).
| Series level of: | Φ3 | Φ2 | Φ1 | ADF | lags | Zt | Result |
|---|---|---|---|---|---|---|---|
| Nominal share prices | 3.24 | 3.29 | 1.65 | 0.20 | 1 | 0.09 | I(1) |
| Real share prices | 1.98 | 1.34 | 6.56** | 2.78 | 4 | 1.80 | I(1) |
| Investment (/K) | 3.31 | 2.22 | 2.37 | 2.17 | 6 | 2.18 | I(1) |
| Cost of capital | 0.99 | 0.77 | 1.04 | 1.33 | 0 | 1.47 | I(1) |
| Real cost of capital | 3.06 | 2.04 | 0.96 | 1.38 | 5 | 1.76 | I(1) |
| Real return on capital | 4.65 | 3.19 | 4.05** | 2.80 | 1 | 2.96** | I(0) |
| Real cash flow | 3.83 | 3.78 | 2.13 | 0.86 | 1 | 0.84 | I(1) |
| Output gap | 11.45** | – | – | 4.79* | 1 | 4.84* | I(0) |
| Real equity raisings | 7.34* | – | – | 3.82* | 1 | 5.75* | I(0) |
| GDP price deflator | 2.92 | 2.84 | 2.49 | 1.53 | 8 | 0.19 | I(1) |
| Oil price | 1.39 | 1.24 | 1.35 | 1.33 | 4 | 1.25 | I(1) |
Notes: The testing strategy recommended by Perron (1988) is followed
where the likelihood ratio tests are: |
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However, visual inspection of corporate investment (as a proportion of the capital stock) and real share prices reveal clear one-off shifts in the series. For the real share-price series there is a downward shift in the 1970s due to the oil price shocks and an upward shift in the late 1980s, explained partly by dividend imputation. For corporate investment (as a proportion of the capital stock) there is a clear shift in the level of the series in 1973.
We follow the methodology suggested by Perron (1989) and redo the unit root tests with the inclusion of a dummy variable to allow for the one-off shifts in the level of the series. The results are presented in Table A2. The appropriate tests statistics to be used in the presence of a break have been calculated by Perron (1989) and are 3.72 and 3.44 at the five per cent and ten per cent significance levels. The results show that real share prices are quite close to being stationary once the one-off shifts in the level of these series are accounted for; corporate investment comes very close to being stationary. As such, corporate investment and real share prices are treated as stationary for the purposes of estimation.
| Series level of: | Φ3 | Φ2 | Φ1 | ADF | lags | Dummy |
|---|---|---|---|---|---|---|
| Real share prices | 4.74 | 3.48 | 8.95* | 3.53** | 4 | 1973:Q1–1979:Q4 |
| Investment (/K) | 5.32 | 3.65 | 5.45** | 3.26** | 6 | 1973:Q1–1995:Q3 |
Note: * and ** denote significance at the 1% and 5% per cent levels, respectively. The test statistics for the ADF tests are from Perron (1989). |
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Footnote
This test involves making non-parametric corrections to the Dickey-Fuller test. Five lags of the residual autocovariance were chosen. [24]