General AR Processes with a Unit Root, and the Augmented Dickey-Fuller Test
The assumption made in Sections 2 and 3 that the data-generating process is an AR(1) process, is not very realistic for macroeconomic time series, because even after differencing most of these time series will still display a fair amount of dependence. Therefore we now consider an AR(p) process:
У і = в0 + X віУч + Uf, Uf ~ iid N(0, a2). (29.37)
ap. Under the unit root hypothesis, i. e. ap = 0 and a0 = 0, the following hold: If model (29.39) is estimated without intercept, then nap ^ (1 – Zp=1 aj)p0 in distribution, where p0 is defined in (29.22). If model (29.39) is estimated with intercept, then nap ^ (1 – Zp- aj)p1 in distribution, where p1 is defined in (29.33). Moreover, under the stationarity hypothesis, plimn^„ap = ap < 0, hence plimn^„ap = -<*>, provided that in the case where the model is estimated without intercept this intercept, a 0, is indeed zero.
Due to the factor 1 – Zp) ap in the limiting distribution of nap under the unit root hypothesis, we cannot use nap directly as a unit root test. However, it can be shown that under the unit root hypothesis this factor can be consistently estimated by 1 – Zp1 a;-, hence we can use nap/|1 – Zp) a;-| as a unit root test statistic, with limiting distribution given by (29.22) or (29.33). The reason for the absolute value is that under the alternative of stationarity the probability limit of 1 – Zp) a may be negative.10
The actual ADF test is based on the t-value of ap, because the factor 1 – Zp1 aj will cancel out in the limiting distribution involved. We will show this for the AR(2) case.
First, it is not too hard to verify from (29.43) through (29.48), and (29.54), that the residual sum of squares RSS of the regression (29.40) satisfies:
RSS =Z U + Op(1). (29.57)
This result carries over to the general AR(p) case, and also holds under the stationarity hypothesis. Moreover, under the unit root hypothesis it follows easily from (29.54) and (29.57) that the OLS standard error, s2, say, of a2 in model (29.40) satisfies:
(RSS/ (n – 3))o-2(1 – a1)2 1 /Wn(x)2 dx – (jWn(x)dx)2
hence it follows from (29.56) that the f-value i2 of 72 in model (29.40) satisfies (29.34). Again, this result carries over to the general AR(p) case:
Theorem 2. ated by (29.39), and let tp be f-value of the OLS
estimator of a p. Under the unit root hypothesis, i. e. a p = 0 and a 0 = 0, the following hold: If model (29.39) is estimated without intercept, then tp ^ t0 in distribution, where t0 is defined in (29.24). If model (29.39) is estimated with intercept, then tp ^ t1 in distribution, where t1 is defined in (29.34). Moreover, under the stationarity hypothesis, plimn^Jp ^jn < 0, hence plim n^Jp = -<*>, provided that in the case where the model is estimated without intercept this intercept, a0, is indeed zero.