Unit Root with Drift vs. Trend Stationarity
Most macroeconomic time series in (log) levels have an upwards sloping pattern. Therefore, if they are (covariance) stationary, then they are stationary around a deterministic trend. If we would conduct the ADF and PP tests in Sections 4 and 5 to a linear trend stationary process, we will likely accept the unit root hypothesis, due to the following. Suppose we conduct the ADF test under the hypothesis p = 1 to the trend stationary process yt = p0 + p1t + ut, where the uts are iid N(0, a2). It is a standard exercise to verify that then plim n^„na 1 = 0, hence the ADF and PP tests in sections 4 and 5 have no power against linear trend stationarity!
Therefore, if one wishes to test the unit root hypothesis against linear trend stationarity, then a trend term should be included in the auxiliary regressions
(29.39) in the ADF case, and in (29.60) in the PP case: Thus the ADF regression
(29.39) now becomes
Ayt = a 0 + I a;-Ayt-1 + apyt-l + ap+1t + ut, ut ~ iid N(0, a2) (29.81)
where the null hypothesis of a unit root with drift corresponds to the hypothesis a p = a p+1 = 0, and the PP regression becomes:
Ayt = a0 + a 1yt-l + a2t + ut, ut = у(L)et, et ~ iid N(0, 1). (29.82)
The asymptotic null distributions of the ADF and PP tests for the case with drift are quite similar to the ADF test without an intercept. The difference is that the Wiener process W(x) is replaced by the de-trended Wiener process:
After some tedious but not too difficult calculations it can be shown that effectively the statistics nap/(1 – X?=1 aj) and tp are asymptotically equivalent to the Dickey-Fuller tests statistics p0 and f0, respectively, applied to de-trended time series.
1 ( W**(1) – 1 N
2 ^/W**(x)2 dx j
1 W**(1) – 1
2 [7 /W **(x)2 dx j
Theorem 5. Let yt be generated by (29.81), and let ap and tp be the OLS estimator and corresponding t-value of ap. Under the unit root with drift hypothesis, i. e. ap = ap+1 = 0, we have nap ^ (1 – XP=1 a;-)p2 and tp ^ t2 in distribution, where
Under the trend stationarity hypothesis, plimn^„ap = ap < 0, hence plimn^„fp
The densities of p2 and t2 (the latter compared with the standard normal density), are displayed in Figures 29.5 and 29.6, respectively. Again, these densities are farther to the left, and heavier left-tailed, than the corresponding densities displayed in Figures 29.1-29.4. The asymptotic 5 percent and 10 percent critical values of the Dickey-Fuller t-test are:
P(t2 < -3.41) = 0.05, P(t2 < -3.13) = 0.10
Moreover, comparing (29.26) with
Р(т2 < -1.64) = 0.77, Р(т2 < -1.28) = 0.89,
we see that the standard normal tests at the 5 percent and 10 percent significance level would reject the correct unit root with drift hypothesis with probabilities of about 0.77 and 0.89, respectively!
A similar result as in Theorem 5 can be derived for the PP test, on the basis of the OLS estimator of a1 in, and the residuals щ. of, the auxiliary regression (29.82):
Theorem 6. (Phillips-Perron test 2) Let ft be the residuals of the OLS regression of yt on t and a constant, and let d2u and be as before, with the й^ the OLS residuals of the auxiliary regression (29.82). Under the unit root with drift hypothesis,
whereas under trend stationarity plimn^„Z2/n < 0.