The purpose in adding the terms Ay, , is to allow for ARMA error processes. But if the MA parameter is large, the AR approximation would be poor unless Zeis large."

After estimating this augmented equation, the tests K(l), /(1), and F(0, 1) discussed earUer are used. These test statistics have been shown to have asymptotically the same distribution as the Dickey-Fuller test statistics and hence the same significance tables can be used.

The Dickey-Fuller (DF) and the augmented Dickey-Fuller (ADF) tests were very popular methods of testing for unit roots. Recently two other test statistics, the Z„ and the Z, test statistics, suggested by PhilHps, are also being commonly used. These statistics are based on a nonparametric modification of the Dickey-Fuller tests and, hence, are beyond the scope of our discussion.

Powers of Unit Root Tests

For many economic time series the DF, ADF, Z„, and Z, tests have consistently shown that the null hypothesis of a unit root (that is a = 1) cannot be rejected. This has led to the conclusion that almost all economic time series are DS type. This result has been attributed to the fact that the tests have very low power against relevant TS (trend stationary) alternatives. They do not reject the hypothesis a = 1, but they do not reject the hypothesis a = 0.95 either.

Another problem that has been pointed out by Choi is that if the errors have a moving-average component, the long autoregressions used in (14.4) to account for serial correlation biases the OLS estimate a of a toward 1, thus suggesting the presence of a unit root, even when a < 1.

Nelson and Plosser (1982) start with equation (14.3) and test a = 1 under the restriction 6 = 0. For the errors e, they assume that

E, = e, + £,„„ < 1, ,- 11D(0, a) (14.5)

after observing the sample autocorrelations of the first differences of U.S. historical data. However, instead of estimating this model, they used (14.4) to account for the moving-average errors. These long autoregressions produce an upward-biased estimate of a, Choi performed a Monte Carlo study assuming that 7 = 5, 6 = 1, a = 0.5 in (14.4), and = 0.3 in (14.5) and found that a from (14.4) is biased toward 1.

What Are the Null and Alternative Hypotheses In Unit Root Tests?

The null hypothesis in unit root tests is H: a = 1. In the theory of testing of hypothesis, the null hypothesis and the alternative are not on the same footing.

"G. W. Schwert, "Tests for Unit Roots: A Monte Carlo Investigation," Journal of Business and Economic Statistics. Vol. 7, 1989, pp. 147-159. His conclusion is that the best test is the ADF with a long k. See, however, the paper by Choi cited in the next footnote, who argues against the long autoregressions.

"I. Choi, "Most U.S. Economic Time Series Do Not Have Unit Roots: Nelson and Plossers (1982) Results Reconsidered," Manuscript, Ohio State University, 1990.

The null hypothesis is on a pedestal and it is rejected only when there is overwhelming evidence against it. That is why one uses a 5% or 1% significance level. If, on the other hand, the null hypothesis and alternative were to be

, is stationary and y, is nonstationary

the conclusions would be quite different. In the Bayesian approach, and H, are on the same footing and hence this asymmetry does not arise.* Tests for unit roots with the null hypothesis being stationary (no unit root) have also been developed and they often give results contrary to those of the unit root tests with the unit root as the null."

In the Box-Jenkins approach, whether or not there is a unit root is decided by visual inspection of the correlogram. If the correlogram tails off, the time series is considered to be stationary. Otherwise, we examine the correlogram of the first differences. The unit root tests are just a formalization of this visual inspection. However, if the unit root tests and the Box-Jenkins approach give conflicting results, it is important to examine why this is so. In practice, it sometimes happens that a unit root test does not reject the null hypothesis of unit root (at the traditional significance levels), although the correlogram tapers off. Hence it is important to examine the correlogram before applying any unit root tests."*

In tests for unit roots it is important to examine the specification of the alternatives. When we say a unit root test has low power, we have to specify for what alternative it has low power. For example, the Phillips test is based on the OLS estimator a of the parameter a in the model y, = ay, , + u„ whereas the PhiUips-Perron test is based on the OLS estimator a of a in the model y, = ix + ay, , + u,. In both cases, the null hypotheses is the same, namely , = ,. The alternative H, is y, = u, in the PhilUps test and y, = p, - , in the Phillips-Perron test.

Very often the alternative to the simple random walk model (a DS model) is

"It is of historical significance that the widely used 5% and 1% levels were originally suggested in a paper by R. A. Fisher in 1926 in the Journal of Ministry of Agriculture in Great Britain. Fisher was a conservative and did not want to change a treatment unless there was overwhelming evidence that the new treatment was better. Fisher is the father of modern statistics and his prescriptions have been followed without any question ever since.

"D. N. DeJong and C. H. Whiteman, "Reconsidering Trends and Random Walks in Macroeconomic Time Series," Journal of Monetary Economics, Vol. 28, No. 2, Oct. 1991, used a Bayesian method and found only two of the Nelson-Plosser series to be DS type. "For tests with the no unit root as null, see J. H. Kahn and M. Ogaki, "A Chi-square Test for a Unit Root," Economic Letters, Vol. 34, 1990. pp. 37-42; J. Y. Park and B. Choi, "A New Approach to Testing for a Unit Root," CAE Working Paper 88-23, Cornell University, 1988; H. Bierens, "Testing Stationarity Against the Unit Root Hypothesis," Manuscript, Free University of Amsterdam, 1990; and D. Kwiatkowski, P. C. B. Phillips, and P. Schmidt, "Testing the Alternative of Stationarity Against the Alternative of a Unit Root: How Sure Are We That Economic Time Series Have a Unit Root?" Econometrics Paper, Michigan State University, November 1990.

*Bierens, "Testing Stationarity," gives an example of the monthly time series on U.S. inflation rate, 1961-1987 (204 observations) where the correlogram tapers off but the Phillips and Phillips-Perron tests do not reject the null of unit root. The test by Bierens for which stationarity is the null does not reject the null of stationarity.

a-* I -

u\l - a) + ct2

If 8 is very large, a 1 under the alternative ,. Hence the test based on a has low power against ,.

The same would be the case where the null hypothesis is a unit root with drift and the alternative is a stationary series around a linear time trend, as considered by Dickey, Bell, and Miller.-" We have

(,: a = I in , = x -f- ay, , + u, ( - 0)

,: < 1 in , - , - = fjL -1- [ ,„, - , - 2( - 1)] + ,

Again we can show that the OLS estimator a from y, = x -f- ay, , -f- , will tend to 1 under , and the test based on a will have lower power under ,. In both these cases the model size under the alternative is larger than under the null, that it, it has more parameters under the alternative than under the null. For this reason Choi suggests the following tests:

Test 1

: a = 1 iny, = ay, , u, ,: <1

: a = 1, x = in , = p, + ay, , + u, ,: a < 1

"Choi, "Most U.S. Economic Time Series." The discussion here is based on that paper.

"D. A. Dickey, W. R. Bell, and R. B. Miller, "Unit Roots in Time Series Models: Tests and

Applications," American Statistician, Vol. 40, 1986, pp. 12-26.

Test 2

a trend-stationary model. In such cases it can be analytically demonstrated that the test has low power. For instance, in the Said and Dickey (1985) paper the alternative to a simple random walk model is a stationary process with a nonzero mean. That is,

.- a = 1 iny, = ay, , + u,

,: a < 1 iny, - 8 = a(y,„, - 8) + ,

In this case, not being able to reject does not mean that there is strong evidence against The test for unit root is based on the OLS estimator a from the equation y, = ay, i + u,. Thus d = 1 1- ]~\- But under the alternative we have, assuming that var(M,) = and that , are serially uncorrelated,

y, = 8 + (1 - aDu,

In large samples, Zyl,/T- 8 + - {1 - a) and Zyy,i/T- 8 + (1 - a). Hence, under ,,

ct\1 - a)