Thus we have h φ 1 or β 0 the alternative

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integrated of order one, difference stationary, or has a unit-root. Thus we have H 0 : φ = 1 or β = 0 . The alternative hypothesis H 1 is that the process is trend-stationary or sta- tionary with constant mean and is given by: H 1 : - 1 < φ < 1 or - 2 < β = φ - 1 < 0 . Thus the unit root test is a one-sided test . The advantage of the second for- mulation of the Dickey-Fuller regression is that the corresponding t-statistic can be readily read off from standard outputs of many computer packages which makes additional computations unnecessary. 7.3.1 The Dickey-Fuller Test (DF-Test) The Dickey-Fuller test comes in two forms. The first one, sometimes called the ρ -test, takes T ( b φ - 1) as the test statistic. As shown previously, this statistic is no longer asymptotically normally distributed. However, it was first tabulated by Fuller and can be found in textbooks like Fuller (1976) or Hamilton (1994a). The second and much more common one relies on the usual t-statistic for the hypothesis φ = 1: t b φ = ( b φ T - 1) / b σ b φ . This test-statistic is also not asymptotically normally distributed. It was for the first time tabulated by Fuller (1976) and can be found, for example, in Hamilton (1994a). Later MacKinnon (1991) presented much more detailed tables where the critical values can be approximated for any sample size T by using interpolation formulas (see also Banerjee et al. (1993)). 7 7 These interpolation formula are now implemented in many software packages, like EVIEWS, to compute the appropriate critical values.
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156 CHAPTER 7. INTEGRATED PROCESSES Table 7.1: The four most important cases for the unit-root test data generating process (null hypothesis) estimated regression (Dickey-Fuller regression) ρ -test: T ( b φ - 1) t-test X t = X t - 1 + Z t X t = φX t - 1 + Z t case 1 case 1 X t = X t - 1 + Z t X t = α + φX t - 1 + Z t case 2 case 2 X t = α + X t - 1 + Z t , α 6 = 0 X t = α + φX t - 1 + Z t N(0,1) X t = α + X t - 1 + Z t X t = α + δt + φX t - 1 + Z t case 4 case 4 The application of the Dickey-Fuller test as well as the Phillips-Perron test is obfuscated by the fact that the asymptotic distribution of the test statistic ( ρ - or t-test) depends on the specification of the deterministic components and on the true data generating process. This implies that depending on whether the Dickey-Fuller regression includes, for example, a constant and/or a time trend and on the nature of the true data generating process one has to use different tables and thus critical values. In the following we will focus on the most common cases listed in table 7.1. In case 1 the Dickey-Fuller regression includes no deterministic compo- nent. Thus, a rejection of the null hypothesis implies that { X t } has to be a mean zero stationary process. This specification is, therefore, only warranted if one can make sure that the data have indeed mean zero. As this is rarely the case, except, for example, when the data are the residuals from a previous regression, 8 case 1 is very uncommon in practice. Thus, if the data do not display a trend, which can be checked by a simple time plot, the Dickey-Fuller regression should include a constant. A rejection of the null hypothesis then implies that { X t }
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