June 27, 1985
Estimation for Autoregressive Processes with Several Unit Roots
Sastry G. Pantula
North Carolina State University
Raleigh, NC 27695-8203
ABSTRACT
p
Let Y satisfy the stochastic difference equation Y = [ a.Y . + e ,
t
t
t
j=l J t-J
for t = 1, 2, ... , where the e
t
are independent and identically distributed
random variables and the initial conditions (Y-p+l' Y_ + 2 ' ... , YO) are fixed
p
...
constants. I t is assumed that the true, but unknown, roots m , m ,
, mp of
l
2
p
[
a.m p - j = 0 satisfy m
= m = 1 and Imjl <' 1 for j = d+l, .•. ,p.
mP
m =
2
l
d
j=l J
We consider a reparameterization of the model for Y that is convenient for
t
...
testing the hypothesis H : m = m = ... = m = 1, and Imjl
d
d
l
2
<
1 for j = d+l, ... ,p.
We consider the likelihood ratio type "F-statistics" to test the hypothesis H .
d
We characterize the asymptotic distributions of the "F-statistics" under various
alternative hypotheses.
Using these asymptotic results, we obtain a sequential
testing criterion that is asymptotically consistent.
Finally, estimated per-
centiles for the distributions of the test statistics are obtained for d
~
5,
by Monte Carlo methods.
1.
Introduction
Let the time series {Y } satisfy
t
p
[ a.Y . + e ,
j=l J t- J
t
t
=
(1.1)
1, 2, ... ,
where {e tIt=l
l~
is a sequence of independent random variables with mean zero and
variance 0 2 •
It is assumed that the e
that E{letI4+O}
< M for
some 0
>
t
are either identically distributed or
0 and all t.
It is further assumed that the
initial conditions (Y-p+l' ... , YO) are known constants.
For the statistics
considered in this paper, without loss of generality we assume that
=
of order p.
Let
o.
0
2 = 1 and
The time series is said to be an autoregressive process
-2-
t
j=l
a. mp-j
=
0
(1. 2)
J
be the characteristic equation of the process.
m , m , •.. ,
1
2
ID
p
The roots of (1.2), denoted by
are the characteristic roots of the process.
Let the observations Y , Y , ... , Y be available.
l
2
n
02
a = (a , a , ... , a )' and
p
1
2
are unknown.
Assume
It is assumed that
The least squares estimator ~ of a
is obtained by regressing Y on Y l' ... , Y
t
tt-p
Lai and Wei (1983) estab1isheq
A
the strong consistency of a under a very general set of conditions.
If Imjl
j = 1, 2, ... , p, then Y converges to a weakly stationary process as t
t
+
<
00.
1,
In
the stationary case, the asymptotic properties of ~ and of related "F-type" 1ikelihood ratio test statistics are well known.
See Mann and Wald (1943), Anderson (1959)
and Hannan and Heyde (1972).
Dickey (1976), Dickey and Fuller (1979), and Fuller (1979)
have considered testing for a single unit root in a p-th order autoregressive process
(i.e., m = 1 and Imjl
1
<
1 for j = 2, 3, ..• , p).
Hasza (1977) and Hasza and Fuller
(1979) considered a p-th order autoregressive process with m = m = 1 and Imjl
1
2
for j = 3,4, ... , p.
<1
Hasza and Fuller (1979) characterized the distribution of
"F-type" statistic for testing m = m = 1.
l
Z
Dickey, Hasza and Fuller (1984) obtained
similar results for testing the unit roots in seasonal time series.
Sen (1985)
considered the distribution of symmetric estimators and that of the estimated
characteristic roots under the assumption m = m = 1. With the exception of
1
Z
Sen (1985), no one has considered the distribution of the test statistics under
the alternative hypotheses.
For a second order process, Sen (1985) studied
the asymptotic distribution of the regression type t-statistic for testing
the hypothesis H : m = 1, Imzl
1
1
<
1, under the assumption HZ: m = m = 1.
1
Z
He observed that the probability of rejecting the hypothesis H , in favor of
1
the hypothesis HO: I m1 1 < 1, Imzl < 1, is higher under HZ than under H . That
1
-3-
iss it i& more likely we concludes incorrectlys that the process is stationary
when there are really two unit roots present.
We propose "F-type" statistics for testing the hypothesis
H : m = m = ..• = m = I and Imjl
d
l
2
d
<
Is j = d+l s .•.
S
p.
We characterize
the asymptotic distribution of the "F-type" statistics under the hypotheses
Hds d = Os Is ... , p.
We use these asymptotic distributions to obtain an
asymptotically consistent sequential procedure to test H versus H - .
d
d l
The pro-
cedure is very general and it successfully answers the question "how many times
should one difference the series to attain stationarity?"
In Section 2 we present the notation and the main results.
In Section 3
we present the estimated percentiles of "F-type" statistics and some possible
We present an example in Section 4.
extensions.
The proofs of the results
are presented in the Appendix.
2.
Notation and Main Results
We consider the following reparameterizat ion of the mode I (1.1),
=
Y
p,t
where Y.
1 st
=
t
i=l
(l-B)i Yt
After expanding
=
(l-B)i
S.Y. 1
1
1-
,t-
1 + e
t
(2.0
,
ith difference of Y ; and B is the backshift operator.
t
in powers of B, if we compare (2.1) with (1.1), we get
(2.2)
for i = Is 2 s ...
(J
p.
S
= TB
Therefore s we can write
+ c
(2.3)
where T .. is an upper triangular matrix with T .. =
1J
1J
C
= (cIs c 2s ...
S
c )' with c. =
p
1
(_l)i-l(~).
1
(_1)i-1(~-11) for j ~ i and
1-
Since the diagonal elements of T
are ±l s T is nonsingu1ar and we can also obtain
B from
(J.
-4-
It is easy to see that a + = a + =
r l
r 2
8r + l = 8r + 2 = ... = 8p = -1.
= a
p
= 0 if and only if
So if one would like to test whether the order of the
process is r instead of p one could use the F-test for testing 8 + = 8 + = ... = 8 = -1
r l
r 2
p
in the regression (2.1).
Our main purpose of the paper, however, is to test the
hypothesis that there are exactly d unit roots.
order p of the process.
So, we assume that we "know" the
(For all our results, all we need to assume is that the
order of the process is less than or equal to p.)
Now, suppose m = m =
l
2
= md = 1 and Imjl
<1
for j = d+1, ... ,p.
Then
we can write (1.1) as,
P
(l-B)d
(l - miB)Y = e
t
t
IT
i=d+1
(2.4)
Define,
Zt = (1-B)d Yt = Yd,t
and
W =
t
K
i=d+1
(l
- miB)Y t .
(2.5)
Then, W is a dth order autoregressive process with d unit roots and Zt is a
t
(p-d)th order process with stationary characteristic roots.
P
IT
i=d+l
Note that,
(1 - ml.0B)Zt = e t
and we can write Zt as
Zt =
t
j=d+1
ej
Z . d + et
t-J-
(2.6)
or as
t
j=d+l n J"
(1_B)j-d-1 z
+
t-l
et
(2.
n
Since Zt is (1-B)d Yt we get,
Yp,t =
j=~+l
n j Yj - l ,t-1 + e t
(2.8)
-5-
Comparing (Z.8) with (Z.O, we get 8
Therefore, m
1
=
...
assumption that Imjl
=
Bdenote
= 8
Z
= ... = 8
m = 1 if and only if 8 = 8
d
1
Z
<
= 0 and 8 =n for j
j
j
d
=
~
d+l.
The
1 for j = d + 1, .•. , p imposes extra conditions on n .
j
< O.
In particular, n + = 8 +
d 1
d 1
Let
1
the ordinary least squares estimator of
YP,t on Y0,t-1' Y1,t-1' ... , Yp-1,t-1·
B obtained
by regressing
Tha t i s ,
(Z.9)
where
... ,
... ,
tt = (Y O,t-1'
= (t , t
1
t
z,
Y
p-1,t-l
t
)
)'
,
n
and
~-~
~
Y( p) = (y p, l' Yp, Z '
Define the "F-statistic" for testing 6
...
F.
~,n
(p)
=
~-,~---
... , Yp,n )
1
= 82
=
=
B
i
=
0 by
-1'"
B(i) C(i) B(i)
i
(MSE)
(Z.10)
n
where
C(i) = (ixi) submatrix consisting of the first i rows and i columns of (t't)
B(i) = (8 1 , 8z '
... ,
8i )'
and
MSE
n
= mean square error of the regression
=
(n-p)
-1
E
t=l
lY
p,t
- B't'Y
t (p)
JZ
We use H to denote the hypothesis
d
H : m = m = ..• = m = 1 and Imjl
d
d
1
Z
We now present the asymptotic properties of F.
~
,n
< 1,
for j = d+1, d+Z, ... ,p.
(p) under H .
d
-1
,
-6-
Theorem 1:
Assume that Y is defined by (1.1).
t
if i
F. (d)
F.
l.,n
~
Then under the hypothesis H ,
d
d
l.
(p)...12...
{
(2.11)
if i
ex>
>d
where
F.(d)
l.
= ~l.
*'
LG(ii)J- l g*
g (i),d
d
(i),d
*
(it).
and the elements of the vector g(i),d and the matrix Gd
and G defined in (A.9).
d
are functl.ons of gd
The elements of gd and G are functions of weighted chid
(See Lemma 4 of the Appendix.)
squares and normal variables.
Notice that the limiting distribution of F. (p) is independent of the order
l., n
p.
Let us denote Fd(d,a) to be the lOO(l-a)% percentile of the distribution of
Since, under H , F. (d) convergence in probability to infinity for i
l.,n
d
> d,
we suggest the following criterion:
Reject H in favor of H _ ' if F. (p)
d
d l
l.,n
i = d, d+l, •.• ,p.
> F.(i,a)
l.
for
Note that,
tim PH LRejecting HdJ = tim PH lF d (p)
,n
n-+<x> . d
n-+<x>
d
... ,
tim PH LF
(p)
d d ,n
n-+<x>
= a
Also, for j
.
>d
p
lRejecting HdJ
tim ·H.
n-+<x>
~
a
=
l.
J
and for j
<d
tim PH. lRejecting HdJ
n-+<x>
J
> Fd(d,a),
F
p,n
(p)
Fd+l,n(P)
> Fp (p,a)J
> Fd(d,a)]
> Fd+l(d+l,a),
-7That iss if we use our sequential procedures asymptotically the chance that
we reject the hypothesis that there are exactly d unit roots in favor of the
hypothesis that there are exactly d-l unit roots is (i) as if there are exactly
d unit roots s (ii) less than or equal to as if there are more than d unit roots
present and (iii) ones if there are less than d unit roots.
In this senses our
procedure is asymptotically consistent.
Considers for examples a second order process.
hypothesis that there is one unit root.
statistic ("equivalently" Flsn(Z»
Suppose we wish to test the
A common practice is to use the t-
for testing 13
1
= O.
Dickey and Fuller (1979)
presented the empirical percentiles for the distribution of the t-statistic s
under the hypothesis HI: m = 1 and Imzl
l
that Imll
Im.1
1
>
< Is
Imzl
<1
1 or m. = -1.)
1.
Note that rejecting HI may mean
(We are excluding the possibility of
However s in practices the rejection of HI is misconstrued
1
as the former.
or m = m = 1.
l
Z
<
This is because "intuitively" we hope that if there are two unit
roots then the test for one unit root will indicate that there is a unit root.
Sen (198S)s using simulation methods s has shown that s a level a test based on
the t-statistic rejected HI more than 100a% of the time when there are in fact
two unit roots.
Our criterion takes into account that exactly d of the roots
are equal to one and the remaining are less than one in modulus and tests the
However s if one is "sure" that there are at most s
hypotheses sequentially.
·unit roots (e.g. s say s = 5) then one may modify the criterion for testing
Hd(d~s) to "reject H
d
in favor of H if Fisn(p)
d l
> Fi(isa)
for i = d s d+ls'''ss.''
In practical applications it is common to include an intercept term in the
model (Z.l)s i.e. s consider
p
y
Pst
=
13
0
+ i=l
E
I3.Y. 1
1
1 1- st-
+
e
t
(Z.lZ)
.
.
Let ~(+)
6
denote t h e or d'111Ci.ry 1 east squares est1mator
0
f 6(+)=(13 •
.13 •
, ....
• I3 )1
0
1
p
-8-
_
obtained by regressing Yp,t on 1, YO,t-l' Yl,t-l' .•. , Yp - l • t - l
regression "F-statistic" for testing 8
(2.12) by
So = 13
1
F~1)
1.n
(p).
Similarly, let
= 8
1
F~2)
1.n
2
= ... = 8
i
Denote the
= 0 in the regression
(p) denote the "F-statistic" for testing
(1)
= ••. = Si = O.
We now present the asymptotic properties of F • (p) and
i n
F(2) (p) under the hypothesis H .
d
i. n
Theorem 2:
Assume that Y is defined by (1.1).
t
F(j)
1.n (p) -
[
F~j)(d)
1
CD
Then, under the hypothesis
d
if i
:;i
if i
>d
for j = 1, 2, where
*
**
and the elements of the vectors b(i),d
and b(i),d
and those of the matrices H(ii)
d,l
.
and H(ii)
,2 are funct10ns
of the vector b and the matrix H* defined in (A.ll).
d
d
d
elements of b
d
and H*
d
The
are functions of weighted chi-square and normal variables.
(See the proof of Theorem 2 in the Appendix.)
Therefore. an alternative criterion is to reject H in favor of H if
d l
d
F~l)
(p)
F~l)
1,n
(p) in the above criterion.
1.n
> F~l)(i,a)
1
for i = d. d+l, ...• p.
One may use
F~2)
1,n
(p) in place of
However, based on the power studies of Hasza and
(1)
Fuller (1979) and Dickey and FUller (1981) we recommend the use of F.
(p) over
1,n
F~2)
1,n
(p).
with So
d-
3.
l
~
From the results of Hasza (1977), it follows that if Y satisfy (2.12)
t
0, then under the hypothesis H •
d
F~~~
(p) converges in distribution to
X2 • where X2 denotes a chi-square random variable with d degrees 0 f f ree d om.
d
d
Simulation for d=S
We first present the elements of G and
S
gs
which will be used in obtaining
the empirical percentiles of the random variables F.(d). F~l)(d) and F~2)(d).
111
From (A.9) and the results of Lemma 1. we get
-9-
QS
1
~
N2
S
Q4
N4 NS - Q4
G =
S
1
~
Symmetric
N2
4
Q3
2
N3NS-~N4
N3 N4 - Q3
N2 NS -N 3N4+Q3
2
N2N4-~N3
1
~
N2
3
Q2
N2 N3 - Q2
1
~
N2
2
Q1
and
2
N1N -N 2N4+ ~N3
S
N1N4 -N 2N3+Q2
2
N N - 1~N2
1 3
gs =
N1N2 - Ql
2
1 (N
~
1 - 1)
where
Q1 = Q*1
Q2 =
Q; - N~ + 2N 2N3 ,
Q3 =
Qi - 1/3 N~ +
2N2N~
Q: - 2/1S N~ - N~ + 2/3 N2N4 17/31S N~ - 1/3 N~ + 4/1S
QS = Q;
Q4 =
~ClO
N = 2 2 E Y .; V.;;
1
i=1 ......
~
ClO
2
N = 2 2 E h.
3
i=1 ~
-
N
2
=
~ClO
2
2
E
i=1
N2 (N S - N;) + 2N 4 NS '
N N +
2 4
2
y. V.
~
;
~
~ClO
3
~(2N4 - N2 )(N S - N;) + 1/3 N2N;
2
4
E l~y. - Y.;JV.;;
i=1
~
......
y.JV. ; N4 = 2
~
~
*
N = 2~ E ll/o Y - y.+y.JV.;N = 2~ E ll/3 y. - Y.;JV.;;
1
i
S
i=1
~
~
~
i=1
~
......
ClO
2
ClO
~
N* = 2 2 E ll/l0 Y2
2
i
i=l
4
-
S
4
1/3 Y + Y:
i
~
~
N* = 2 2 E l 1/42 Y2 - lIs Y4 + 3 7
Yi
3
i
i
i=1
ClO
2.
Q~ =
E y. J V~ , j ?; 1
J
~
i=1 ~
1
2
ClO
2y? JV. ;
~
ClO
-
~
9
oy. J V. ;
~
~
5
-10-
and
Yi == 2j(2i_l)~j-l(_1)i+l
.
Also,
H* ==
S
and
h
S
==
"5 ]
[:5
G
S
[::]
where
and
!.:
N == t"
6
OD
L
i==l
ll/24 y1 - O.S y~ + y?jVo .
...
~
~
~
All of the F and the F(j) statistics can be computed using the elements of
H* and h
S
S·
Estimates of the percentiles of the F-statistics are presented in Tables 3.1,
3.2 and 3.3.
For the finite sample sizes the F-statistics were computed for
samples generated using the model (1.1), with
et~NID(O,1)
and YO == Y- == ... ==Y:. ==0.
4
l
To estimate the percentiles of the limiting distributions of the test statistics
we use the method of simulation employed by Dickey (1976) and Hasza (1977).
Briefly, the method consists of approximating the sequence {Yi}~==l by a finite
sequence.
Using the Monte Carlo methods sample distribution functions of the
statistics are generated by appropriately truncating the infinite series in the
d e f ~nitions
o
0
f N'
. s, N*'
. s an d Q*'
os.
J
J
J
The percentiles were "smoothed" by fitting
a regression function to the original set of estimates.
-ll-
The estimated standard errors of the estimated percentiles are generally
less than 0.90 percent of the table entry for the limiting distributions and
are generally less than 1.50 percent of the table entry for the finite sample
cases.
Fifty thousand indpendent sample statistics were used in constructing
the percentiles for the asymptotic distribution.
In our discussion. we assumed that the initial conditions are fixed.
The
limiting distribution does not depend on the initial condition (Y-p+l •...• YO).
but these values will influence the small sample distribution.
criterion we proposed in Section 2 for testing H
d
F
d .n
(d.a) where F
statistic F
Fd(d.a).
F.
~.n
(p)
d .n
d .n
(d).
involved Fd(d.a) rather than
(d.a) denotes the 100(1-a)% empirical percentile of the
For small samples. one should use F
> F.
~.n
(i.a) for i
=
d. d+l •...• p.
1
and 8
2
Compute the regression F-statistics F
for testing the hypotheses. (i) 8
=
(i,a).
~.n
Consider. for example. a second order autoregressive process {Y }. Suppose
t
2
SO. We regress Y
2 .t = (I-B) Yt on YO .t- 1 = Yt- 1 and Y1 .t- 1 = (I-B)Y t- l'
to obtain 8
a
(d.a) instead of
The asymptotic results of Section 2
~
=
d .n
Therefore. our criterion is to reject H in favor of H if
d
d l
are still valid even though we replace F.(i.a) by F.
n
Also. the
0.10.
1
=0
and (ii) 8
1
=
8
2
1 .n
= O.
(2) and F
2 •n
respectively.
(2)
Let
Hasza and Fuller (1979) suggest that one should reject H : two unit
2
roots (in favor of HI: exactly one unit root) if F • (2)
2 n
>
2.82.
Also. a
common practice is to reject HI in favor of stationarity if F • (2)
l n
>
3.01.
Our criterion differs from above in the sense we reject HI in favor of
H : no unit roots. only if F
(2)
O
l .n
the condition F
2 .n
(2)
>
> 3.01
and F
2 ,n
(2)
> 2.82.
If we do not add
2.82. then with probability greather than 0.10 we will
be concluding that the process is stationary when in fact it has two unit roots.
However, if we are absolutely sure that there is at most one unit root then we
may use the criterion of rejecting HI in favor of H if F • (2)
O
l n
>
3.01.
-12TABLE 3.1
Empirical Percentiles for the F-Statistics
F
1, n
(1)
n
0.50
0.80
0.90
0.95
0.975
0.99
25
50
100
250
500
0.58
0.59
0.60
0.60
0.60
0.61
1.89
1.89
1.89
1.89
1.89
1.88
3.04
3.01
2.99
2.98
2.97
2.96
4.34
4.23
4.18
4.15
4.14
4.13
5.74
5.54
5.42
5.35
5.32
5.28
7.80
7.38
7.16
7.02
6.97
6.91
0.95
0.97
0.98
0.98
0.98
0.99
2.04
2.02
2.02
2.01
2.01
2.01
2.88
2.82
2.79
2.77
2.76
2.75
3.76
3.62
3.55
3.50
3.49
3.47
4.71
4.45
4.32
4.24
4.22
4.19
5.98
5.59
5.38
5.23
5.17
5.10
1.15
1.18
1.19
1. 20
1. 20
1.20
2.22
2.20
2.19
2.18
2.18
2.17
2.97
2.88
2.83
2.81
2.80
2.80
3.73
3.55
3.46
3.41
3.39
3.39
4.49
4.21
4.07
3.99
3.96
3.94
5.57
5.n
4.88
4.75
4.70
4.66
1. 29
1.32
1.34
1.35
1.35
1.35
2.35
2.31
2.29
2.28
2.28
2.28
3.07
2.95
2.89
2.86
2.85
2.84
3.80
3.56
3.45
3.39
3.37
3.35
4.56
4.17
3.99
3.88
3.85
3.84
5.60
4.97
4.67
4.51
4.46
4.46
1.37
1.41
1.43
1.44
1.45
1.45
2.44
2.38
2.36
2.34
2.34
2.34
3.17
3.02
2.94
2.90
2.88
2.87
3.90
3.60
3.46
3.38
3.36
3.36
4.64
4.16
3.95
3.84
3.81
3.83
5.68
4.93
4.58
4.40
4.36
4.38
00
F
2,n
(2)
25
50
100
250
500
00
F
3,n
(3)
25
50
100
250
500
CD
F
4,n
(4)
25
50
100
250
500
CD
F
5,n
(5 )
25
50
100
250
500
00
-13-
Table 3.2
Statistic
F(l)(l)
1, n
n
0.50
0.80
0.90
0.95
25
50
100
250
500
2.36
2.41
2.43
2.44
2.45
2.45
4.99
4.94
4.91
4.91
4.91
4.91
6.95
6.74
6.65
6.60
6.58
6.58
8.96
8.54
8.35
8.24
8.24
8.21
10.98
10.36
10.04
9.84
9.78
9.69
13.84
12.76
12.24
11.93
11.83
11. 76
2.55
2.56
2.57
2.58
2.58
2.59
4.43
4.30
4.24
4.21
4.20
4.19
5.79
5.49
5.34
5.25
5.23
5.20
7.15
6.60
6.35
6.22
6.18
6.15
8.56
7.69
7.33
7.14
7.09
7.06
10.51
9.14
8.59
8.33
8.27
8.23
2.68
2.67
2.67
2.67
2.67
2.67
4.39
4.19
4.08
4.02
4.01
3.99
5.56
5.16
4.96
4.85
4.81
4.79
6.78
6.ll
5.78
5.60
5.55
5.52
8.00
7.03
6.56
6.30
6.22
6.19
9.68
8.23
7.54
7.17
7.07
7.06
2.80
2.76
2.74
2.73
2.73
2.72
4.51
4.20
4.05
3.97
3.94
3.93
5.67
5.ll
4.84
4.69
4.65
4.63
6.83
5.96
5.55
5.33
5.27
5.26
8.05
6.74
6.20
5.95
5.88
5.84
9.76
7.83
7.06
7.70
6.61
6.55
2.40
2.34
2.32
2.31
2.30
2.30
3.87
3.51
3.37
3.29
3.28
3.26
4.87
4.23
3.98
3.86
3.84
3.82
5.90
4.92
4.54
4.36
4.32
4.29
6.91
5.60
5.08
4.83
4.77
4.73
8.39
6.46
5.73
5.41
5.34
5.29
00
F(l)(2)
2,n
25
50
100
250
500
00
F(l)(J)
3,n
25
50
100
250
500
00
F(l)(4)
4,n
25
50
100
250
500
00
F(l)(5)
5,n
Empirical Percentiles of the F(l)-statistics
25
50
100
iso
500
00
0.975
0.99
-14-
Table 3.3
Statistic
F(2) 0)
1,n
Empirical Percentiles of the F
0.50
n
25
50
100
250
500
00
0.80
0.90
(2) S
.
.
- tat1st1cs
0.95
0.975
0.99
1.71
1.72
1.72
1. 72
1. 72
1. 72
3.09
3.00
2.96
2.94
2.94
2.94
4.12
3.94
3.85
3.80
3.79
3.78
5.16
4.87
4.72
4.64
4.61
4.58
6.29
5.81
5.57
5.44
5.39
5.36
7.77
7.02
6.66
6.46
6.40
6.37
2.05
2.04
2.03
2.03
2.03
2.03
3.34
3.20
3.13
3.10
3.09
3.08
4.26
3.99
3.86
3.79
3.76
3.75
5.20
4.75
4.54
4.43
4.40
4.38
6.18
5.50
5.20
5.06
5.02
4.99
7.56
6.48
6.06
5.86
5.82
5.78
2.30
2.26
2.25
2.24
2.23
2.23
3.61
3.40
3.30
3.24
3.Z2
3.21
4.52
4.14
3.96
3.86
3.82
3.80
5.46
4.86
4.58
4.42
4.37
4.36
6.43
5.57
5.17
4.94
4.88
4.86
7.73
6.50
5.92
5.61
5.52
5.51
2.49
2.43
2.40
2.38
2.37
Z.37
3.89
3.59
3.44
3.36
3.33
3.31
4.86
4.32
4.07
3.93
3.89
3.87
5.80
5.02
4.65
4.44
4.39
4.38
6.81
5.65
5.18
4.95
4.89
4.85
8.Z6
6.55
5.87
5.55
5.47
5.41
2.63
2.53
2.49
2.47
2.47
2.46
4.14
3.72
3.54
3.45
3.43
3.41
5.18
4.44
4.15
4.02
3.99
3.97
6.23
5.14
4.71
4.52
4.48
4.44
7.34
5.83
4.25
4.99
4.93
4.89
8.87
6.71
5.91
5.57
5.49
5.44
.. _.---
.~--_._._-
F(2)(2)
2,n
25
50
100
250
500
00
F(2)O)
3,n
25
50
100
250
500
00
F(Z)(4)
4,n
25
50
100
250
500
00
F(2)(5)
5,n
25
50
100
250
500
00
-15-
In addition, we also computed F.
1,n
F.
1,n
(d,a) are not included.)
(d,a) for 1 ~ i
<d
~
5.
(Tables for
It is interesting to note that the empirical
percentiles F.
(d,a) increase in d for a fixed i and a ~ 0.5.
For example,
Fl,lOO(l, 0.05)
= 4.18,
=
F ,100(4, 0.05)
1
= 6.77
1,n
F 1 ,100(2, 0.05)
5.04, F l ,100(3, 0.05)
and F ,100(5, 0.05)
l
= 7.14.
6.20,
This indicates that if one
uses the criterion, "reject the hypothesis that there is exactly one unit root
in favor of stationarity if F
1 ,n
> F 1 ,n (l ,a),"
(p)
then, with probability greater
than a, one would conclude that the process is stationary when really there are
several unit roots.
Except for some tedious bookkeeping, the results can be extended to regressions involving time trend in the model.
4.
Example
Pankratz (1983) lists 70 consecutive monthly volume of commercial bank real
estate loans, in billions of dollars, starting from January 1973.
By fitting a
sixth order autoregressive process we concluded that a third order autoregressive
process provides an adequate fit for the data.
Y
3,t
=
(1-B)3 y
t
= y
t
-
3Y
t-l
+ 3Y
. t-2
- Y
on Y
Y
t-3
t-l'
l,t-l
and Y ,i-l = (1-B)2 Yt _ = Y - 2Y 2
t 1
1
t 2
with
Y ,t
3
=
;2
0.0839.
=
Regressing
+
(l-B)Y
t-1
= Yt-l
- Y
t-2
Y - , we get
t 3
0.00139Y t _ l - 0.1045Y 1 ,t_l - 1.3061Y 2 ,t_l
(0.00094)
(0.0795)
(0.1233)
+
et
(4.1)
The numbers in the parentheses are estimated standard errors
calculated by the usual regression formulas used in most computer regression
routines.
From (4.1) we calculated the F-statistics for testing the relevant hypotheses.
The calculated values are F ,70(3)
l
=
2.19, F ,70(3)
2
= 1.19
and F ,70(3)
3
= 47.29,
where F ,70(3) is the regression F-statistic for testing the hypothesis that
i
-16-
the first i coefficients of (4.1) are zero.
Since 47.Z9 is greater than any
of the table values for F ,50(3) and F ,100(3), we reject the null hypothesis
3
3
that there are three unit roots.
Note that, under the hypothesis HZ:
there are two unit roots and the third
root is less than one in absolute value, the statistics F
asymptotically equivalent.
with F ,50(2, 0.20)
z
= 2.02
and F ,100(2, 0.20)
Z
autoregressive model for YZ,t
with
+
(3) and F
Z ,n
Comparing the calculated value of F ,70(3)
Z
hypothesis H at ZO% level of significance.
2
(1
Z ,n
=
= 2.02,
(Z) are
=
1.19
we fail to reject the
Finally, we fitted a first order
(1-B)2 Yt to obtain
0.36B)(1-B)Zy
(O.lZ)
t
02 = 0.0842.
Same conclusions were obtained when an intercept was included
in the model (4.1).
ACKNOWLEDGEMENTS
I wish to express my thanks to Professor D. Boos, D. Dickey and W. Fuller
for their helpful comments.
I am very grateful to Dr. Dickey and Dr. Sen for
letting me use their Monte Carlo programs.
-17-
APPENDIX
We prove Theorem 1
in several steps.
autoregressive process with d unit roots.
distributions of F
i ,n
First we consider a dth order
For this process we obtain the limiting
(d), i = 1, 2, ... , d.
process with d unit roots.
Then extend the results to a pth order
Define,
( A.l)
=
So ,t =
... ,
for i = 1, 2,
t
I: S. 1
.
j=l 1- ,J
et
d and t = 1, 2,
... ,
where e
t
is as defined in Section l.
Note
that,
(i-B) j S.
= S ..
1,t
1-J,t
for j
~
i and S.
is an ith order autoregressive process with i unit roots.
1,t
Consider W = Sd,t'
t
a dth order autoregressive process with d unit roots.
In
the parameterization of (2.1), we can write W as
t
Wd,t
=
W.1,t
=
d
I:
i=l
8. W
1
i-l,t-l
+ et
(A.2)
where
and 8
1
= 8
2
= ... = 8
d
= O.
Let
B denote
the ordinary least squares estimator of
B in the regression of Wdt on Wa,t-l' Wl,t-l'
-=
B
... ,
W
d-l,t-l
-1
b
B
d,n d,n
where the (ij)th element of B
is
d,n
n-l
n-l
W.1- 1 ,t W.J- 1 ,t = I: Sd-i+l,t Sd-j+l,t
t=l
t=l
I:
That is,
(A.3)
-18-
and the ith coordinate of b
d,n
is
n-1
n-1
1:
W. 1
Wd
1 = t~l Sd-i+1,t SO,t+1 .
t=l ~- ,t
,t+
.
We now obtain the orders in probability of the elements of B a n d b
d,n
d ,n
We
also obtain recursive relationships between the elements.
be as defined in (A. 1).
Lemma 1:
Let S.
(i)
= l(i-l)!j-l
~,t
S.
~,t
(ii) n
-(i-~)
S.
~
,n
.J4
~
(t-r+l)(t-r+2) .•. (t-r+i-l)e
r=l
-
N.
~
~
~
(iv)
n-l
i
1: S. t
= 0 (n )
SO,t+l
~,
P
t=l
i
~
(v)
n-l
+
1: S. t Si-l,t = ~ S2
i,n-l
t=l ~,
o
Proof:
as n
~ w.
1,
1,
p
(n U - 2 ), i ;;: 2,
1:
- n-l
t=l
Si-l,t Si-k+l,t +
The results (i) and (ii) are straightforward.
2i
El
E
t=l
° (n 2i ),
P
n
i j
= 0 (n + ).
1: S.
S.
~,t
J,t
P
t=l
Also,
n
Vlt~l Sit SO,t+l j =
and we get (iv).
n
t
1:
1:
t=l k=l
Therefore,
n
=
1: S:
t=l ~,t
and hence
~
n-l
1: S. t
= S.
1 Si-k+l,n-l
~,nt=l ~, Si-k,t
for 2 ~ k ~ i.
n-
Now, for i
~
r
N(O,o~), o~ = l(2i-l){(i-l)!}2j-1
n-l
i j
(iii)
= 0 (n + ) for i, j
1: S. t S.
~,
J,t
P
t=l
(vi)
Then,
2,
1
1
n
n
Now,
o
p
(n 2i - k- 1 ),
-19-
n-1 2
1: 8.
t=l l.,t
and hence
n-1
1: 8.l.,t- 1 8 l.-,t
. 1
t=l
=~
8
2
i,n-1
n-1 2
- ~ 1: 8.
t=l l.-l,t
Therefore,
n-1
1:
t=l
n-1
n-1 2
= 1: 8.
+ 1: 8. 1
18 . 1
l.,t 8.l. - 1 , t
t=l l.,t- l.-,t t=l l.- , t
8.
2
2i 2
= ~ 8
+ 0 (n - )
i,n-1
p
Note that, for 2 ;:;; k
n-1
~
i,
n-1
= 1: 8.
8. 8. k
1 8.l. - k , t +
t=l l.,t l.- ,t t=l l.,t1:
o
(n 2i - k - 1 )
P
Now,
n-1
n-1
n-1
8
8
8
t:1 8 i ,t-1 i-k+1,t = t:1 8 i ,t-1 i-k+1,t-1 + t:1 8 i ,t-1 i-k,t
and hence
n-1
n-1
n-1
8
t:1 8 i ,t-1 8 i - k+ 1 ,t-1 = t:1 8 i ,t-1 i-k+1,t - t:1 8 i ,t-1 8 i - k ,t
Also,
n-1
n-1
n-1
8
t:1 8 i ,t 8 i - k+1 ,t = t:1 8 i ,t-1 8 i - k+1 ,t + t:1 8 i - 1 ,t i-k+1,t
and hence
n-1
n-1
8.l.,n- 1 8.l.- k+ 1 ,n- 1 = t=l
1: 8.l.-,t
1
8 i-k+ 1 ,t + t=l
1: 8.l.,t- 1 8.l.-,t
k
Therefore,
n-1
n-1
L S.l.-,t
1
8 i-k+,t
1
t:1 8 i ,t-1 8 i - k ,t = s.l.,n- 1 8.l.- k+ 1 ,n- 1 - t=l
From Lemma 1, it follows that, for
~ ~
j
~
i and i
~
u
2,
n-1
k-1 r
+ (_l)k n-1
8.
1: 8.
1 = 1:(-1) 8.
18 .+ +1
1
1:
t=l
l.,t- 1 J,tr=O
l.-r,n- J r ,nt=l
2
.s i-k,t-1
i f i-j is even
-20-
=
k-l
r
k 2
E(-l) 8.~-r,n- 1 8 J.+r +1 ,n- 1 + (-1) ~8.~- k ,n- 1
r=O
+ 0 (ni+j-l),if i-j is odd,
P
where k = the integer part of %(i-j).
-1
We now obtain the order in probability results for B
d ,n
d
d-l
Leuuna 2: Let»
= diag {n , n
, ... , n}. Define,
d,n
G
d,n
= »-1
B
= »-1
b
d,n
»-1
d,n
d,n
and
d,n
d,n
Then,
=
0 (1)
p
= 0 (1)
p
and the determinant of G
is bounded away from 0 in probability.
dn
From Leuuna 1, it is clear that G
= 0 (1) and g
= 0 (1).
p
dn
p
dn
Proof:
the result on determinants through induction on d.
G
l,n
= n
E 8
t=l
~
and since n
-2 n
n
-3
8
We prove
For d = 1,
2
l,t
2
2,n
-3 2
8
convergence in distribution to a constant times a chi-square
2 ,n
random variable with one degree of freedom, we get G
is bounded away from zero.
l ,n
Assume now that the det
Amax (Gd - 1 ,n )
~ trace
lG~:l,n]
is bounded in probability.
lG 1 ] = 0 (1).
d - ,n
p
-1
Also, since we assumed that det lG - ,n]
d 1
-1
is bounded in probability we get A . (G
m~n
l
show that det lG- ] is 0 (1).
d,n
p
Note that
d-
Note that
1
,n
) is bounded in probability.
We now
-Zl-
... ,
n
-1 n-l
r
t=l
S
d,t
,
Sl).
,t
Let O(n) denote
the vector of regression coefficients in the regression of Sdt on
=
c'
t
(n-(d-U S
d-l,t' ... , n
-1
Then,
Sl,t) .
-1
O(n) = Gd-l,n~
and
detlG
d ,n
]
Note that
=
n
-(Zd-Z)
4
+ 0 (n Zd - l ) .
~Sd,n-l
p
1
-(Zd-l)SZ
2
2
·
S ~nce
n
d,n-l converges in distribution to ad Xl ' we get
n-
Zd
11<1~ liZ is bounded away from zero.
Also
Also,
A-
Z
max
lG
d-l,n
JII qn lIZ;;;
Therefore,
"O(n) "
and
110(n),,-1
Now, for n*
=
< n,
* - O(n)J ' G _ n* lO(n * ) - O(n)]
n -Zd lO(n)
d l,
-ZZ-
We now show that n€
> 0,
for n
find M
1
> N1 .
> 0,
M
Z
Zd
>
lIlO(n*)1I - 110(n)IIJ
Z
is bounded away from zero.
Given
° and N1 large such that
Let,
=
n*
integer part
and
N
Then, for n
=
> N,
integer part
> N1
n*
and
M
PlIIO(n*)1I
>
Therefore, for n
< 2"1
d
n J
>1
- €.
N,
M
Pl{lIO(n)1I - lIo(n*)IIJZ
> ~
nZdJ
>1
- Z€.
Since A . lG 1 *J is bounded away from zero, we get detlG
J is bounded away from
d ,n
m1n d -,n
U
zero in probability.
We now obtain the asymptotic distributions of G
and gd .
d,n
,n
A
n
where L
= L'L
n n
(A .4)
is an (n-1)x(n-1) lower triangular matrix with every element below and on
n
the diagonal is equal to 1.
L' = II'
n
where 1
Define,
=
Then,
+ I n-1 - Ln
(1,1, ... ,1)': (n-1)x1, I
Let Al
of A and let x.
n
1n
with A. .
1n
= (x.1n l'
(A. 5)
n-
lis an identity matrix of size (n-l).
>A
, n ,Zn
> ... >
x.1n Z,.··,x.1n,n- 1)'
A 1
denote the eigenvalues
n- ,n
denote the eigenvector
Define the orthogonal transformation of e
by
u.
1n
=
n-1
tEl x int e t · ,
Define
n
into
U
n
=
(u
ln
associated
,u
zn , •.• ,un- 1 ,n )'
-Z3-
Hasza and Fuller (1979) established that, for a fixed k,
Following the arguments of Hasza and Fuller (1979) it can be shown that
n- Zk e' Ak e
n
n
V
~
n
~
i=l
yZk vZ
i
-i
= Q*
k
l)~J-l (_l)i+l and {v.}
where y. = ZL(Zi 1
is a sequence of normal independent
1
(0,1) random variables.
Also, the random variables N defined in Lemma 1 may be
k
ClO
written as N = i~l 0kiVi for some constants {Oki}.
k
distribution of the diagonal elements of Cd
Lemma 3:
~
for k
n
Let N. ( ) = n
1, n
-(i-Yz)
Now we obtain the limiting
,n
S.
l' where S.
1 is defined in (A. 1).
1,n1,n-
1,
-Zk
k
A e
n
n
=
k
I:
i=l
a
(k)
.
(A. b)
n,1
)'
h
"th 1
of a(k)..
where S n(Zk)= (S Zk,l' S Zk,Z' ••• , S
Zk,n-l ' t e J __ e ement
n,1 1S
n
Then,
-Yz Z(k-i)
v
I:
t.=O
Proof:
. Z(k-i)-t
(~)
k,i,t n
l
+ O(n- ) and v
k,i,t
We prove this using induction on k.
n
-Z A
0
f
h f
t e orm
is a finite positive constant.
For k =1, from Dickey (1977),
e
n n
and hence (3.b) holds.
holds for k + 1.
Now, we assume that (A.b) holds for k and show that it also
Note that,
n -(Zk+l) L Ak e
n n n
= n -1
=
k
I:
i=l
k
I:
L
i=l
n
a
(k)
.
n,1
(k)
(_l)kn-(Zk+l)s(Zk+l)+ 0 (n- l )
f n,i
NZi,(n-l) +
n
p
where
f(k~ = n -1 L
n,1
n
a
(k)
n,i
Note that the jth element of f(k~
n,1
is
(A.7)
-24-
n
= n
j
L
-1
r=l
a
(k)
.
n,1.,r
-k 2(k-i)+1
2
L
£.=0
where v*
k, i,£.
is a constant.
Now,
n-(2k+2) Ak+ 1 e =
n
n
= n- 1 l11' +
= n-
1
~
i=l
I
k
- L J L A e
n
n n n
n-1
lIl' + I
n-
1 - L Jf(k~ N
n n,1. 2i,(n-1)
where
a(k:1) = n-lll(I'f(k~) + f(k~ - L f(k~J
n,1.
n,1.
n,1.
n n,1.
Note that the jth element of a
a
(k+l)
..
n,1.,J
= n -k2
i
=
1, 2, ... , k.
(k+1)
.
has the form,
n,1.
2(k-i)+2
. 2(k-i)+2-£'
-1
L
v
(.1)
+ O(n ).
£.=0
k+1,i,l n
Therefore,
n
-(2k+2)
A e
n n
(k+l)
= k+1
L a . N2 · (
i=l
n,1.
1., n- 1) +
(_1)k+1 S (2k+2)
n
0 (n-l) .
+ p
u
Note, from (A. 0) and (A. 7) we get
2k
82
n -4k n-1
L
=n -4k e 'A e
t=l 2k,t
n n n
k
- 2(-1)
k
k
E
k
L
i=l j=l
i~l N2i ,(n-1)n
+ Op(n- 1 ),
(k)'
8
(k) N
n ,i an,j
N
2i,(n-1) 2j,(n-l)
-2k a(k). 'S(2k)
n,1. n
-Z5-
and
n
-(4k+Z) n-l 2
L S
t=l 2k+l,t
= n -(4k+Z),
e
n
A
~
- Z(-l)k
i=1
+0 (n- l
p
2k+l
e
n
n
N
n-(2k+l)f(k~'s(Zk+l)
Zi,(n-l)
n,1 n
>.
It can be shown that,
n
-r-~ n-l
_
L (!)m S
r,t
t=l n
for any integers m, r
~
O.
V
N*
~ N(O,o*Z )
m,r
m,r
In fact, one can express
CD
N*
= L 0*
V.
m,r i=l m,r,i 1
where V.
1
~
NID(O,l).
n- 2k a(~)IS(2k)
n1
n
Therefore,
=n
-2k n-1
L
t=1
= 2(k-i)
v
L
l=o
a
(k)
S
.
n,1,t 2k,t
k,i,l
n
-Zk-~ n-1
+ 0 (n-1)
t 2(k-i)-l
L (-)
S·
t=1 n
2k,t
-
v
p
N(a) , say.
i,k
Similarly,
n -(Zk+l) f(k)' s(Zk+l)
n, i
n
.L
Z(k-i)+1
E
l=o
*
vk,i,l
N*
2(k-i)-l+l,2k
=
N( f)
i,k
Also, there exist finite positive constants a~ . k and f~ . k such that
1,J,
1,J,
.(..01.m a (k)'
. a (k).
n~
n,1 n,1
= a *. .
1,J,
k
and
Therefore,
Q*2k -
k
L
k
L
i=1 j=1
*
a . . k N21·N ·
2J
1,J,
-Z6-
~ N 1. N
+ Z(_l)k+l
Z
i=l
0
(a)
i, k
and
n
V
-(4k+Z) n-l Z
1:
S
t=l 2k+l,t
k
k
= QZk+l
1:
1: f~ ° k NZiN
Q
*
- i=l
Zk
Zj
j=l 1.,),
k l
+ Z( _U +
Now, using the results of Lemma 1, we get for 0
-(i+j) n-l
V
1: S 1.,t.
1 S)., t -1 t=l
n
~
j
k
1:
i=l
~
N( f)
N
Zi i,k
i and i
~
2,
P d ,1.,)
°
•
where
k-l
1: (_l)r No
N)o+r+l +
r=O
1.-r
k
(-1)
Qk' if i-j is even
Pd,i,j =
k-l
1: (_l)r N.
N)o+r+l + (_l)k ~N7 k ' if i-j is odd,
r=O
1.-r
1.and k = integer part of
-1 n-l
1: S
n
t=l
~(i-j).
Note also that
V
Z
SO,t+l~(Nl
- U .
l,t
Therefore,
V
G
d,n -
Gd
and
<A. 9)
where G and gd consist of the corresponding limiting variables.
d
We now present
the limiting distributions of the "F-type" statistics.
Lemma 4:
Let F. (d) be as defined in (Z.1O) with Y = W ·
t
1., n
t
F o (d)L 1..-1 g(*'i), d
1.,n
(ii)
Where G d
[G~iil]
-1
*
g( i), d = F.1. (d)
is the i x i upper left hand submatrix of G~
-1
1
and
vector consisting of the first i elements of G gd . (Note that
d
Then, for i
~
d,
-27
Proof:
Recall,
F. (d) =
1.,n
B(i )
l i( MS E)n J - 1
= l i(MSE) J
n
-1
B'U~1.
-1
-
C(i) fi( i)
-1 U~J-l
U~ 6
lU~ B
1.
1. d,n 1.
where
U(i) = (i x i) left hand corner submatrix of I d
and
6 = B- 1
b
d,n
Note that,
as defined in (A.3).
(MSE)
n
d,n
= (n-d)
= (n-d)
P
~
-1
-1
n
2
W
d,t
t=l
E
n
E
t=l
e
- 6'
b
d,n
l
0 (n- )
t + P
2
l.
Also,
and
D
n
B- 1
d,n
D
n
...E...
d d-l
, ... ,n}.
where D = diagonal {n ,n
n
F.
1., n
(d)
Therefore,
--v
lJ
Now we present the proof of the theorem.
Proof of Theorem 1:
Recall that Y is a pth order process with d unit roots.
t
Let W and Zt be as defined in (2.5).
t
Then, W is a dth order autoregressive
t
process with d unit roots and Zt is a (p - d)th order autoregressive process with
stationary roots.
We can express Zt as,
-28-
P
Zp-d,t = j=~+l
=
where Z.
1,t
(1 - B)
i
Zt'
nj
Zj-d-l,t-l + e t
Note that n +
d l
< 0.
From Fuller (1976), it follows
that
N(O , T-lr-Zl T- l ,)
k - - 11) - V
n-"'(11
where 11
(A. 10)
is the least squares estimator of 11 obtained by regressing Z
on
p-d,t
ZO,t-l' ... , Zp-d-l,t-l; T is given in (2.3); the (i,j)th element of the matrix
Let
A
B
be obtained by regressing
i
Note that, for
Yp,t on Yo,t-l' Y1 ,t-l' ..• , Yp- 1 ,t - l' where Y.1, t = (1- B) Yt •
i ~ d, Y.
= (l_B)i-d Z
1, t
t
= Z. d
1-
,t
.
Therefore,
Bis
obtained by regressing
Let us
Zp-d,t on YO,t-l' Y1,t-l' ... , Yd - l ,t-1' ZO,t-l' Zl,t-1' ... , Zp-d-l,t-1'
partition
B as
(B(1)"
n')'.
A
We can obtain 11 using several regressions.
First,
Let
regress Zp-d,t on .~ = (YO,t-l' Yl,t-l'
A(p-S)
R
= Z
- 41' 11
, be the tth residual from this regression where
p-d,t
p-d,t
t
A(p-S)
11
is the corresponding least squares estimator of 11. Similarly, regress
Zi,t-l on
R
p-d,t
4I~
on R
For i
and obtain Ri,t and n(i), for i = 0, 1, ... , p-d-l.
Finally, regress
O,t' Rl , t' ... , Rp-d-l,t' to obtain 11.
< d,
let S*d .
-1,t
= s*i,t
Then,
= Y.1, t
= l(i-l)IJ
-1
t
E
r=l
(t-r+l)(t-r+2) ... (t-r+i-l)Z
r
Using Kronecker's lemma and the arguments of Lemma 1, it can be shown that,
E
t=l
and
y2
i,t
-29-
n
~
l.
t=l
for 0
~
i,j
~
2d
Y•
Y.
1= 0 P (n - i -j)
1,t- 1 J,t-
d-1.
n
I:
t=l
Similarly, one can show that,
=
Y.1,t- 1 Zt- k
and hence
n
I:
t=l
for i = 0, 1, 2, ... , d-1;
Now, consider, for 0
n
Also, for 0
~
~
-(2d-i-j)
i
=
Y.
1 Z
1
1,tj,t-
~
~
i
j
k
= 0,
~
d-1,
1, ... , p; j
n
= n -(2d-i-j)
t=l W.1,t- 1 W.J,t- 1
I:
n
I:
t=l
l Yp-d+i,t - nd+1 Y i,t-1
d-1,
=-
n d+1 n
-(d-i)
n
-1
-1
-1
-1
= I: D
111'111 D
lilt 1fJ't Dn
n
n
n
t=l
D
-2
nd+ 1
G
d,n
~ Y
t=l 1,t- 1
Therefore,
=
0, 1, ... , p-d.
1
+ 0 (n- )
p
l.
•
Z
d +1
p-,t
+ 0
P
(n-
1)
-
-30-
and, we get
... (i)
Dn T'I
=
= 0 p (1)
D
n
"'(p-d)
=
T'I
i = 0,1, ... , p-d-l
-n d+l
G
-1
l
0 (n- )
d,n gd,n + p
= 0 (1)
p
where D = diagonal {n
n
n
~
t=l
R.
=
R.
1,t J,t
=
d
,
n
d-l
... ,
,
n
-
~
t=l lZi,t-l
n
.'
t
n} .
Now, note that, for
... (i)j
T'I
lz
~
Z.1,t- lZ,J,t- 1 +
R
d
t=l
j,t-l
°
~
i, j:ilp-d-l,
- .'n{j)]
t
0 (1) .
P
Similarly,
n
n
~
t=l
R.
1,t
p- ,t
= t=l
L
Z.
1 Z d
+ 0 P (1).
1,tp-,t
Therefore, we get,
T'I -
T'I
and
-
where B is defined in (A.3), and
F.
1,n
(p)
= F~1,n (d) +
T'I
is defined in (A.lO).
Also, for i
V
F. ( p ) - F.(d).
1,n
1
d,
l
0 (n- )
p
where F~ (d) is the ifF-statistic" for testing 13 = 13 = ... = 13. =
1
1,n
2
1
regression of Wd,t on WO,t-l'
~
Wl,t-l' ... , Wd-l,t-l.
Therefore,
° in the
-31-
Now, consider,
=
Fd+1 ,n(P)
1
(d+1)(MSE)
n
where C(d+1) is the (d+1) x (d+1) submatrix consisting of the first (d+1) rows
and (d+1) columns of the matrix,
n
E
t=l ·t~~
An
=
n
E
t=l
and
~I
~t
= (2
!.:
" " " ,n 2} "
O,t-1'
2
n
~t.tl
E
t=l ~t~~
... ,
1,t-I'
-1
2
p-d-1,t-1 )"
d d-l
~
Let, D* = diagonal {n ,n
,""",n,n,
n
Then,
D* A
n
n
Therefore, under the assumption of H : exactly d unit roots
d
-
2
-1
nd+1 L( d+ 1) w* J
p
Therefore,
_
under Hd "
V
Fd+i,n -
v
00
Now, since Fd+i,n(P)
00
,
under Hd "
2, we have that
u
-32-
We now outline the proof of Theorem 2.
Proof of Theorem 2:
(1,
.~,
Let
,,(+)
~
be obtained by regressing
t' ) where ·t and tt be as defined
t
n
n
E
E
n
t=l t
t=l
n
n
n
A(1)=
E
E
E
n
t=l
t=l ·t t=l ·t·~
n
n
n
E
E tt
E
tt·~
t=l
t=l
t=l
.'
n(1)=
D
and
1
diagonal
n
T
(1)
=
n
E
e
E
.'e
t t
E
t'e
t t
t=l
n
t=l
n
t=l
{n~ ,
n
d
,
n
d-l
,
Y
P,t
on
in the proof of Theorem l.
Define,
-1
t'
t
·tt~
ttt~
... ,
n, n ~ ,
... ,
1
n~t
,
t
Then, it follows from
Theorem 1 that
0'
r- 1 .]
and
(A.ll )
where
H*
d
=
~d
Pd = ( Nd+l' Nd ' ... , N2)' ,
~
= diagonal (1,
-n d+ l , -n d+ l ,
..• ,
-n d+ l )
-33-
and the matrix G and the vector gd are as defined in Theorem 1.
d
Now, following
the arguments used in proving Theorem 1, we can show that, under M ,
d
F~ 1) (d)
F~1)()...R-.
1,n p
{
1
,
00
i
:;ii
d
i
>d
where
F~1)( ) = the F-statistic for testing 8 =
1,n p
1
= 8 = 0 in the
i
regression of (2.12),
*1
lH(ii)J- l *
F~ 1) (d) = i1 h(i),d
h( i), d
1
d,l
H(ii) = (i x i) submatrix consisting of the rows 2 to
d,l
columns 2 to
i+l of the matrix H*-1
d
i+l
and
h(i),d
= i x 1 vector consisting of the elements 2 to
*
i+l
of the
and
vector H*-1 h .
d
d
Similarly,
F~2)( ) _
1,n p
F~2) (d)
V
i
~
d
1
{
00
, i
>d
where
F~2)( ) =
1,n p
the F-statistic for testing 80 = 8 =
1
= 8. = 0
1
in the regression of (2.12)
F~2)(d) =
1
1
**'
i+l h(i),d
lH(ii)J- l **
h( i),d
d,2
H(i,i)= (i+l) x (i+l) submatrix consisting of the first (i+l) rows
d,2
and (i+l) columns of the matrix H*-1 '
d
and
(i+l) x 1 vector consisting of the first (i+l) elements of
u
-34-
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