In the discussion of the ADF test we have assumed that the lag length p of the auxiliary regression (29.81) is fixed. It should be noted that we may choose p as a function of the length n of the time series involved, similarly to the truncation width of the Newey-West estimator of the long-run variance in the Phillips – Perron test. See Said and Dickey (1984).
We have seen that the ADF and Phillips-Perron tests for a unit root against stationarity around a constant have almost no power if the correct alternative is linear trend stationarity. However, the same may apply to the tests discussed in Section 6 if the alternative is trend stationarity with a broken trend. See Perron (1988, 1989, 1990), Perron and Vogelsang (1992), and Zivot and Andrews (1992), among others.
All the tests discussed so far have the unit root as the null hypothesis, and (trend) stationarity as the alternative. However, it is also possible to test the other
way around. See Bierens and Guo (1993), and Kwiatkowski et al. (1992). The latter test is known as the KPSS test.
Finally, note that the ADF and Phillips-Perron tests can easily be conducted by various econometric software packages, for example TSP, EViews, RATS, and
for arbitrary є > 0.
This density is actually a kernel estimate of the density of p0 on the basis of 10,000 replications of a Gaussian random walk yt = yt-l + e, t = 0, 1,. . ., 1,000, yt = 0 for t < 0. The kernel involved is the standard normal density, and the bandwidth h = c. s10,000-1/5, where s is the sample standard error, and c = 1. The scale factor c has been chosen by experimenting with various values. The value c = 1 is about the smallest one for which the kernel estimate remains a smooth curve; for smaller values of c the kernel estimate becomes wobbly. The densities of p1, t1, p2, and t2 in Figures 29.2-29.6 have been constructed in the same way, with c = 1.
To see this, write 1 – Xp=1PjPJ’ = ПР=1(1 – PjL), so that 1 – XjUPj = Пір=1(1 – Pj), where the 1/Pjs are the roots of the lag polynomial involved. If root 1/pj is real valued, then the stationarity condition implies -1 < Pj < 1, so that 1 – Pj > 0. If some roots are complexvalued, then these roots come in complex-conjugate pairs, say 1/p1 = a + i. b and 1/p2 = a – i. b, hence (1 – p1)(1 – p2) = (1/p1 – 1)(1/p2 – 1)p1p2 = ((a – 1)2 + b2)/(a2 + b2) > 0.
In the sequel we shall suppress the statement "without drift." A unit root process is from now on by default a unit root without drift process, except if otherwise indicated.
10 For example, let p = 2 in (29.37) and (29.39). Then a1 = – pl7 hence if P1 < -1 then 1 – a1 < 0. In order to show that P1 < -1 can be compatible with stationarity, assume that P1 = 4P2, so that the lag polynomial 1 – P1L – P2L2 has two common roots -2/| p1|. Then the AR(2) process involved is stationary for -2 < P1 < -1.
11 The most important difference with other econometric software packages is that EasyReg is free. See footnote 1. EasyReg also contains my own unit root tests, Bierens (1993, 1997), Bierens and Guo (1993), and the KPSS test.
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