733 unit root test testing strategy independently

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7.3.3 Unit-Root Test: Testing Strategy Independently whether the Dickey-Fuller or the Phillips-Perron test is used, the specification of the deterministic component is important and can pose a problem in practice. On the one hand, if the deterministic part is underrep- resented, for example when only a constant, but no time trend is used, the test results are biased in favor of the null hypothesis, if the data do indeed have a trend. On the other hand, if too many deterministic components are used, the power of the test is reduced. It is therefore advisable to examine a plot of the series in order to check whether a long run trend is visible or not. In some circumstances economic reasoning may help in this regard. Sometimes, however, it is difficult to make an appropriate choice a pri- ori. We therefore propose the following testing strategy based on Elder and Kennedy (2001). X t has a long-run trend: As X t grows in the long-run, the Dickey-Fuller regression X t = α + δt + φX t - 1 + Z t should contain a linear trend. 10 In this case either φ = 1, δ = 0 and α 6 = 0 (unit root case) or φ < 1 with δ 6 = 0 (trend stationary case). We can then test the joint null hypothesis H 0 : φ = 1 and δ = 0 by a corresponding F-test. Note that the F-statistic, like the t-test, is not distributed according to the F-distribution. If the test does not reject the null, we conclude that { X t } is a unit root process with drift or equivalently a difference-stationary (integrated) process. If the F-test rejects the null hypothesis, there are three possible situations: (i) The possibility φ < 1 and δ = 0 contradicts the primary observa- tion that { X t } has a trend and can therefore be eliminated. 10 In case of the ADF-test additional regressors, ∆ X t - j , j > 0, might be necessary.
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160 CHAPTER 7. INTEGRATED PROCESSES (ii) The possibility φ = 1 and δ 6 = 0 can also be excluded because this would imply that { X t } has a quadratic trend, which is unrealistic. (iii) The possibility φ < 1 and δ 6 = 0 represents the only valid alterna- tive. It implies that { X t } is stationary around a linear trend, i.e. that { X t } is trend-stationary. Similar conclusions can be reached if, instead of the F-test, a t-test is used to test the null hypothesis H 0 : φ = 1 against the alternative H 1 : φ < 1. Thereby a non-rejection of H 0 is interpreted that δ = 0. If, however, the null hypothesis H 0 is rejected, this implies that δ 6 = 0, because { X t } exhibits a long-run trend. The F-test is more powerful than the t-test. The t-test, however, is a one-sided test, which has the advantage that it actually corresponds to the primary objective of the test. In Monte-Carlo simulations the t-test has proven to be marginally superior to the F-test. X t has no long-run trend: In this case δ = 0 and the Dickey-Fuller re- gression should be run without a trend: 11 X t = α + φX t - 1 + Z t .
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