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univariate time series. The limiting distribution of the ADF test for
p
1was
derived by Dickey and Fuller [7], and it can be shown that this distribution is the
same for
k
≤
k
−
>
1 and for
k
=
1. Fuller tabulated the approximate critical values for the
ADF test with
k
1for
specific
sample sizes.
Finite-sample critical values for the ADF test for
any
sample size were obtained
by means of response surface analysis by MacKinnon [18], who also showed that an
approximate asymptotic distribution function for the test can be derived via response
surface estimation of quantiles [19].
Although the asymptotic distribution of the ADF test statistic does not depend on
the lag order, it is noted by Cheung et al. [5] that empirical applications must deal
with finite samples, in which case the distribution of the ADF test statistic can be
sensitive to the lag order. Taking this into account, they closely examined the roles
of the sample size and the lag order in finding the finite-sample critical values of the
ADF test.
As we noted above, the limiting distribution of the ADF test statistic is the same
for
k
≥
1 and
p
≤
k
−
>
1 and
k
=
1. Hence, for simplicity, we consider the case of
k
=
1. In fact, let
Y
denote the autoregressive AR(1) model:
(
)=
(
−
)+
ε
,
=
,
,...,
Y
t
cY
t
1
t
1
2
(18.6)
t
2
where
Y
(
0
)=
0,
c
is a real number, and
ε
t
∼
N
(
0
,
σ
)
(i.e.,
ε
t
is normally distributed
2
with zero mean and variance
).
From the AR(1) model (18.6), one can see that the condition
c
σ
for all
t
=
1
,
2
,...
=
1 in (18.6) is
equivalent
to
the
requirement
that
the
reverse
characteristic
polynomial
det
z
of AR(1) has a unit root. In other words, to determine whether
an autoregressive time series AR(1) has a unit root, we must test the null hypothesis
H
0
:
c
(
1
−
cz
)=
1
−
1.
Let
y
1
,
=
y
2
,...,
y
T
denote a sample of
T
consecutive observations of the AR(1)
process
Y
(
t
)
, then the maximum likelihood estimator of
c
is the least squares esti-
mator:
T
t
=
1
y
t
y
t
−
1
∑
=
∑
1
y
t
−
1
.
c
(18.7)
t
=
Note that
c
is a consistent estimator of the regression coefficient
c
.
Then the ADF statistic is given by
1
T
T
t
1
y
t
−
1
ε
t
∑
=
T
(
c
−
c
)=
1
y
t
−
1
.
(18.8)
1
T
2
t
∑
=
Dickey and Fuller [7] derived the following representation of the limiting distribu-
tion for statistic
T
(
c
−
c
)
:
1
2
Γ
−
1
W
2
T
(
c
−
c
)
⇒
(
−
1
)
,
as
T
→
∞
,
(18.9)
