TimeSeriesBook.pdf

Thus we have either φ 1 and α 0 or φ 1 and α 6 0

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Thus we have either φ = 1 and α = 0 or φ < 1 and α 6 = 0. The null hypothesis in this case therefore is H 0 : φ = 1 and α = 0 . A rejection of the null hypothesis can be interpreted in three alternative ways: (i) The case φ < 1 and α = 0 can be eliminated because it implies that { X t } would have a mean of zero which is unrealistic for most economic time series. (ii) The case φ = 1 und α 6 = 0 can equally be eliminated because it implies that { X t } has a long-run trend which contradicts our primary assumption. (iii) The case φ < 1 and α 6 = 0 is the only realistic alternative. It implies that the time series is stationary around a constant mean given by α 1 - φ . 11 In case of the ADF-test additional regressors, ∆ X t - j , j > 0, might be necessary.
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7.3. UNIT-ROOT TESTS 161 As before one can use, instead of a F-test, a t-test of the null hypothesis H 0 : φ = 1 against the alternative hypothesis H 1 : φ < 1. If the null hypothesis is not rejected, we interpret this to imply that α = 0. If, however, the null hypothesis H 0 is rejected, we conclude that α 6 = 0. Similarly, Monte-Carlo simulations have proven that the t-test is superior to the F-test. The trend behavior of X t is uncertain: This situation poses the follow- ing problem. Should the data exhibit a trend, but the Dickey-Fuller regression contains no trend, then the test is biased in favor of the null hypothesis. If the data have no trend, but the Dickey-Fuller regression contains a trend, the power of the test is reduced. In such a situa- tion one can adapt a two-stage strategy. Estimate the Dickey-Fuller regression with a linear trend: X t = α + δt + φX t - 1 + Z t . Use the t-test to test the null hypothesis H 0 : φ = 1 against the al- ternative hypothesis H 1 : φ < 1. If H 0 is not rejected, we conclude the process has a unit root with or without drift. The presence of a drift can then be investigated by a simple regression of ∆ X t against a constant followed by a simple t-test of the null hypothesis that the constant is zero against the alternative hypothesis that the constant is nonzero. As ∆ X t is stationary, the usual critical values can be used. 12 If the t-test rejects the null hypothesis H 0 , we conclude that there is no unit root. The trend behavior can then be investigated by a simple t-test of the hypothesis H 0 : δ = 0. In this test the usual critical values can be used as { X t } is already viewed as being stationary. 7.3.4 Examples for unit root tests As our first example, we examine the logged real GDP for Switzerland, ln(BIP t ), where we have adjusted the series for seasonality by taking a moving-average. The corresponding data are plotted in Figure 1.3. As is evident from this plot, this variable exhibits a clear trend so that the Dickey- Fuller regression should include a constant and a linear time trend. Moreover, { ∆ ln(BIP t ) } is typically highly autocorrelated which makes an autoregres- sive correction necessary. One way to make this correction is by augmenting 12 Eventually, one must correct the corresponding standard deviation by taking the auto- correlation in the residual into account. This can be done by using the long-run variance.
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