Concluding Remarks

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






The useful comments of three referees are gratefully acknowledged.

See Chapter 26 on spurious regression in this volume. This phenomenon can easily be demonstrated by using my free software package EasyReg, which is downloadable from website http://econ. la. psu. edu/~hbierens/EASYREG. HTM (Click on "Tools", and then on "Teaching tools").

See Chapter 30 on cointegration in this volume.

The reason for changing the subscript of a from 1 in (29.1) to 0 is to indicate the number of other parameters at the right-hand side of the equation. See also (29.39). Recall that the notation op(an), with an a deterministic sequence, stands for a sequence of random variables or vectors xn, say, such that plimn^„xn/an = 0, and that the notation Op(an) stands for a sequence of random variables or vectors xn such that xn/an is stochastically bounded: Ve Є (0, 1) 3 M Є (0, ~): supn>1P(| xn/an | > M) < є. Also, recall that convergence in distribution implies stochastic boundedness.

The Borel sets in R are the members of the smallest o-algebra containing the collec­tion ©, say, of all half-open intervals (-», x], x Є R. Equivalently, we may also define the Borel sets as the members of the smallest o-algebra containing the collection of open subsets of R. A collection 9 of subsets of a set Q is called a o-algebra if the following three conditions hold: Q Є 9; A Є 9 implies that its complement also belongs to 9: QA Є 9 (hence, the empty set ф belongs to 9); An Є 9, n = 1, 2, 3,. .., implies Un=1 An Є 9. The smallest o-algebra containing a collection © of sets is the intersection of all o-algebras containing the collection ©.

Under the assumption that et is iid N(0, 1),

„ n











P(max1<t<n | et | < є ) =


1 – 2




for arbitrary є > 0.

Подпись: 7Подпись: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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Bierens, H. J. (1997). Testing the unit root hypothesis against nonlinear trend stationarity, with an application to the price level and interest rate in the U. S. Journal of Econometrics 81, 29-64.

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Dickey, D. A., and W. A. Fuller (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057-72.

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Perron, P. (1989). The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361-402.

Perron, P. (1990). Testing the unit root in a time series with a changing mean. Journal of Business and Economic Statistics 8, 153-62.

Perron, P., and T. J. Vogelsang (1992). Nonstationarity and level shifts with an application to purchasing power parity. Journal of Business and Economic Statistics 10, 301-20.

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Said, S. E., and D. A. Dickey (1984). Testing for unit roots in autoregressive-moving average of unknown order. Biometrika 71, 599-607.

Zivot, E., and D. W.K. Andrews (1992). Further evidence on the great crash, the oil price shock, and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251-70.

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