Ju 1Y 13 ~ 1988
Asymptotic Distributions of the Unit Root Tests
When the Process is Nearly Stationary
Abbreviated Title:
Unit Root Tests
Sastry G. Pantula *
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203 - USA
*
Partially supported by the National Science Foundation under grant NSF-DMS-8610127.
Key Words and Phrases: Random walk, white noise, unit root tests, limiting
distributions, autoregressive and moving average processes.
AMS 1980 Subject Classifications:
Primary 62M10.
Asymptotic Distributions of Unit Root Tests When
the Process is Nearly Stationary
by
Sastry G. Pantu1a
North Carolina State University
Raleigh, NC 27695-8203
ABSTRACT
Several test criteria are available for testing the hypothesis that the
autoregressive polynomial of an autoregressive moving average process has a
single unit root.
Schwert (1988), using a Monte Carlo study, investigates the
performance of some of the available test criteria.
He concludes that the
actual levels of the test criteria considered in his study are far from the
specified levels when the moving average polynomial also has a root close to
one.
This paper studies the asymptotic null distribution of the test
statistics for testing p
approaches one.
=
1 in the model Y
t
=
pY - + e - ee _ , as e
t 1
t 1
t
It is shown that the test statistics differ from one another
in their asymptotic properties depending on the rate at which e converges to
one.
1.
INTRODUCTION
Fuller (1976) and Dickey and Fuller (1979) have proposed simple tests for
testing the hypothesis that the characteristic equation of an autoregressive
process has a single unit root.
The test criteria proposed by Dickey and
Fuller (1979), however, are not valid when the process is an autoregressive
moving average process.
The moving average parameters in the model introduce
some bias into the Dickey-Fuller test statistics, which are constructed
assuming that the model is a pure autoregressive process.
Said and Dickey
(1984, 1985) propose two methods for testing for a single unit root in an
autoregressive moving average process.
Phillips (1987a) and Phillips and
Perron (1988) suggest alternate criteria, which approximately correct the bias
in the Dickey-Fuller test statistics.
Hall (1988), on the other hand, suggests
a criterion based on the instrumental variable approach which avoids the bias
in the Dickey:Fuller criteria.
Schwert (1988) and Hall (1988) consider the model
( 1.1)
where e
t
is a sequence of iid N(O,1) variables.
Schwert (1988) cites several
examples of economic data that satisfy model (1.1) with a close to one.
also Schwert (1987).
See
Schwert (1988) considers the test statistics, for testing
the null hypothesis p=1, proposed by Dickey and Fuller (1979), Said and Dickey
(1984, 1985), Phillips (1987a) and Phillips and Perron (1988).
Using a Monte
Carlo study, Schwert (1988) compares the empirical level with the specified
level of different test criteria.
He concludes that the performance of the
-2-
test criteria is poor when e is close to one.
= 0.8,
When e
the empirical
levels of different test criteria are observed to be much higher than the
specified level.
Hall (1988) observes that his criteria based on the
instrumental variable approach also reject the null hypothesis more often than
expected, when e is close to unity.
The main goal of this paper is to study the asymptotic properties of the
test criteria when e approaches unity.
and e
Note that the model (1.1) with p
=1
= 1 reduces to Yt = e t , a white noise process. The null hypothesis
Ho : p=1, lei < 1, includes processes that are arbitrarily close to a white
noise process.
If the process were a white noise process, one should reject
the null hypothesis that the process is nonstationary (H : p=1, lei < 1).
O
In
finite samples, it may not be reasonable to expect that the above test criteria
hold the proper level for all values of e, especially when e is arbitrarily
close to one.
To study the behavior of the test statistics when e is close to
one, we consider the model (1.1) with p
and 0
~
o.
If 0
= 0, we get e =
invertible moving average process.
=1
and e = e
n
=
1-n
-0
~,
where 0 <
~
- ~ and the process is an integrated
As 0 increases, e gets closer to one and
the process behaves like a white noise process.
In section 2 we discuss the model and the unit root test statistics.
In
section 3 we present the asymptotic distributions of the test statistics for
different values of
o.
We conclude with some remarks in section 4.
proofs of the main results are given in the mathematical appendix.
The
< 2
-3-
2.
THE MODEL AND THE TEST STATISTICS
Consider the simple model
(2.1)
where p = 1, e
-0
n
= 1 - n Y, 0 < Y < 2, 0
variables with mean zero, variance
0
2
~
0 and e t is a sequence of iid random
and bounded fourth moments.
Without loss
of generality, we assume throughout the paper that a 2 = 1, eo = 0 and Yo = o.
Note that,
=et
+ n
-0
(2.2)
Y X t 1
t
where Xt = Ei=lei is a random walk process.
S6, the process Yt(n) is a
combination of a white noise process and a random walk process.
If 0 = 0, then
Yt(n) is a nonstationary process given by Y = Y - + e - (1-Y)e _ .
t
t 1
t
t 1
As 0
increases to infinity, Yt(n) behaves more and more like a white noise process.
(In fact, l'f o~ > 0.5 then
max1~t~n
n-olX t-l I converges t 0 zero a 1mos t sure 1
y.)
In this sense, the process is nearly stationary (in fact, nearly white noise).
The model (2.1) is motivated by the examples of economic data cited in Schwert
(1988) .
Ahtola and Tiao (1984), Phillips (1987b) and Chan and Wei (1987) consider an
analogous specification of the autoregressive coefficient.
processes that are nearly nonstationary, where p = p
increases.
n
They consider
converges to one as n
The purpose of their studies is to investigate the asymptotic power
-4-
of the tests under a sequence of local alternatives.
We, on the other hand,
are considering a sequence of models in the null hypothesis that approach the
alternative (stationary models) and study the asymptotic level of the tests.
Now, let us consider the behavior of the sampleautocovariance function
,),(h) of the Yt(n) process given in (2.1).
n
E
Yt(n) \-h (n)
t=h+1
n
-20+1
-1
= n
E ete t _h + n
t=h+1
Note that
,),(h) = n -1
')' 2n-2
n
2
-0
E
Xt - 1 + 0P (n )
t=h+1
Therefore, if 0 < 0 < 0.5,
n20 - 1 Y(h)~
y2 r , h=0,1,2, ...
1
where r =
o
f
2
W (t) dt and W(t) is a standard Brownian motion.
Therefore, for
o
< 0 <.0.5 ,
(We use the notation "~,, to indicate the covergence in distribution and "~,,
to denote the convergence in probability.
We will also use the notation
,,~"
to signify weak convergence of the probability measures on the function space
0[0,1], where 0[0,1] is the space of real valued functions on the interval
[0,1) that are right continuous and have finite limits.)
o
> 0.5, then
Y(h)~
0
and
A
P
')'(0)- 1 .
for h > 0
On the other hand, if
-5-
In fact, if 0 > 0.75, then from Corollary 6.3.5.1 of Fuller (1976) it follows
that, for any fixed h,
n
where I
~-level
1/2
A
[p(1),
B
A
... , p(h)]- N(O, I
h
)
denotes an identity matrix of size h.
h
Therefore, when 0 > 0.75, an
test criterion based on the Q-statistic suggested by Ljung and Box
(1978) is expected to reject the hypothesis that the process is white noise
only 1000% of the time.
That is, when 0 > 0.75, in practice one would tend to
believe that the process is white noise.
A limited simulation of the processes
Yt(n) in (2.1) with 0 > 0.75, indicates that the time plot and the sample
correlogram resemble that of a white noise process.
(Is it really so bad that
we conclude the process is white noise when 0 > 0.75?)
We now present some of the test statistics that exist in the literature
for testing the null hypothesis H : p = 1, lei < 1 in the model (1.1) and
O
present the asymptotic distributions under the null hypothesis (i.e., p=1 and
0=0
in (2.1)).
1.
T statistic suggested by Dickey and Fuller (1979):
The regression t-statistic for testing the coefficient of Y - 1 is zero
t
in the regression of Y on Yt - 1 ,
t
(2.3)
where
2
s1
= (n-2)
-1 n
E (Y
t=2
and
t
- p Y - )
t 1
2
-6-
Phillips (1987a) has shown that, under the model (2.1) with p = 1 and 0 = 0,
(2.4)
where
1
2
and W(t) is a standard Brownian motion.
Note that if, in addition, 7=1
(2.5)
Fuller (1976) gives the percentiles of the limiting distribution given
in (2.5).
2.
t
k
statistic suggested by Said and Dickey (1984):
The regression t-statistic for testing the coefficient of Y - is zero
t 1
in the regression of ut = Yt - Yt - 1 on Yt - 1 , ut _ 1 ' ... , ut - k + 1 ' where
k = 0(n 1/ 4 ). That is,
(2.6)
where
~1
is the first coordinate of
~
-1
= Gk 9 k '
-7-
u t - 1 ' ... , u t - k +') ,
= (Yt-1'
2
sk = (n-2k)
-1
n
A
E (u t t=k+1
Lt-1~)
-1
and Gil is the (1,1)th element of G .
k
2
Said and Dickey (1984) have shown
that under the model (2.1) with p=1 and &=0,
t
3.
t
GN
E
r-1/2e.
k.,
statistic proposed by Said and Dickey (1985):
Consider the one-step Gauss Newton estimator (p,a)' obtained by
.
minimizing En =1[e
(p,a)] 2 , starting with an initial estimator (p *,a *)',
t
t
1 = 0 (n- 3 / 4 ) and a* - a = 0 (n- 1/ 4 ). (Said and Dickey (1985)
where p*
P
p
suggest using p* = 1 and
a*
= Durbin's estimator of
a.
In Result
A.1 of the appendix, we show that it is not necessary to take p* = 1.
We
3 4
-1/4).
show that it is sufficient to have p* - 1 = 0 (n- / ) and a* - a = o
( n
p
p
In fact, we show that, as long as the initial estimators are properly
chosen, the iterated Gauss Newton estimator may be used to test p=1,
instead of the one-step Gauss Newton estimator with p*=1.)
Then, under the
model (2.1) with p=1 and &=0, the t-statistic
t GN
= ( S2e C 1 )-1/2(p- -1)
(2.7)
-8-
where
2
se
n
- - 2
E [et(p,a)] ,
t=1
= n-1
* * is the
Ft(p,a)
2x1 vector of partial derivatives of et(p,a) with respect to p and a.
4.
Zq statistic proposed by Phillips (1987a):
let p denote the least squares estimator in the regression of Y on
t
Yt - 1 .
Then, under the model (2.1) with p=1 and 0=0, the statistic
= n (p-l)
Z
q
fJ
-r
-1
- (2 n
-2 n 2
-1 2
E Yt -1 )
(s k
t=2
n,
(2.8)
(2.9)
~,
where
s
2
= n-1
u
s
n
2
t=l
t
E u
2
2
-1 k
=
s + 2n
E
n,k
u
. 1
J=
and k = 0(n 1/ 4 ).
The percentiles of the limiting distribution (2.9) of Zq
are tabulated in Table 8.5.1 of Fuller (1976).
5.
ZIV statistic suggested by Hall (1988):
Let PIV denote the instrumental variable estimator of p in the
regression of Y on Y - where Yt - 2 is used as an instrument.
t
t 1
actually suggests a class of estimators where
instrument.)
Yt-~' ~>1,
(Hall (1988)
is used as an
Then, under the model (2.1) with p=1 and 0=0, the statistic
-9-
(2.10)
In the literature, there are several other test statistics (e.g., t
k ,}J.
and
t k ,or of Said and Dickey (1984), Zt' Za.p. , Zt}J.' Zerr and Zt or of Phillips and
Perron (1988) and the instrumental variable versions of Phillips and Perron
(1988) test statistics proposed by Hall (1988».
The purpose of the paper
is to give some indication of what happens to some of the test statistics when
the process is nearly stationary.
The ideas presented here can be routinely
extended to other test statistics and hence will not be presented here.
3.
MAIN RESULTS
For different values of 0 in model (2.1), we now present the asymptotic
distribution of the five test statistics described in Section 2.
-0
under model (2.1), Yt(n) = e + n ')' \-1' where \
t
earl ier, if
o
-0
> 0.5, then n
max1~t~nIXt_11
t
= E i =1 e i .
Recall that
As noted
converges to zero almost surely.
This indicates that the behavior of Yt(n) (and hence of the statistics) may
depend on whether 0 > 0.5 or not.
Note also that the test statistics t k and ZQ depend on k which is of order
n1/4 .
In fact, the t-statistic t
k
is based on approximating the integrated
moving average processes in (1.1) by a kth order autoregressive process.
(2.1), we have
From
-10-
= -(1-n
-0
Y)e _
t
k-1
= -
Y) u
~=1
The remainder term (l-n
not converge to zero.
+ e
t
~
-0
E (l-n
1
t-~
+
e
t
- (l-n
-0
k
Y) e _
t k
(3.1)
-0
k
Y) e t - k in the autoregressive approximation mayor may
Since k=0(n 1/ 4 ), the coefficient (1_n- oy)k converges to
zero or one depending upon 0 < 0 < 0.25 or 0 > 0.25.
Therefore, in presenting
the limiting distributions of the test statistics we consider three intervals:
(i)
0 < 0 < 0.25,
(ii) 0.25 < 0 < 0.5 and (iii) 0 > 0.5 .
3.1
Results for the Case 0 < 0 < 0.25:
"Nonstationary Region"
The asymptotic distributions of the five statistics are summarized in the
following theorem.
Theorem 3.1:
Consider the model (2.1) with p=1 and 0 < 0 < 0.25.
the initial estimators p* and e* used in the computation of t
* - 1 = 0p (n- 1 ) and e* - en = 0p (n-0.5-0.50).
p
JJ
(b)
t
(c)
t
(d)
Z
(e)
ZIV
k
---+
-
JJ
GN
0:
-
JJ
-
JJ
Then,
r-1/2~
r-1/2~
r
r
-1
-1
~
,
~
,
where the test statistics, r and
~
are defined in Section 2.
GN
Assume that
are such that
-11-
Notice that the test statistic
~
diverges to negative infinity.
More
importantly, note that the remaining statistics have the same asymptotic
distributions as in the case 0=0.
That is, even if the process is nearly
stationary, the test statistics have the asymptotic distributions as in the
case when the process is nonstationary, provided the rate at which the moving
= 1-nOy
average parameter en
4
converges to one is not faster than n 1/ .
In this
sense, the test criteria based on t k , t GN , Zq and ZIV are robust and are
expected to retain the proper level for 0 < 0 < 0.25.
3.2
Results for the case 0.25 < 0 < 0.5: "Grey Zone"
The following theorem summarizes the limiting distributions of the test
statistics given in Section 2 for processes satisfying (2.1) with p=1 and
0.25 < 0 < 0.5.
Theorem 3.2:
Consider the model (2.1) with p=1 and 0.25 < 0 < 0.5.
L
_[y2r ] -1/2 ,
(b)
n -ok t
(c)
-U+O.5
zq -13
n
(d)
n
k
-U+0.5
Then
[~r(1 (N
,
3 - N2 )
13
ZIV--+ [y2r ]-1(N - N ) ,
1
0
where NO' N , N and N are independent N(O,1) random variables independent of
2
3
1
r and
~.
-12-
Notice that the test statistics
~
and t
k
diverge to negative infinity.
The asymptotic distribution (properly normalized) of l C( is the same as that of
20-0.5
Note that IlC(1 and 'lIV' diverge to infinity at the rate n
, whereas
. . t d'lverges to negatlve
.
. f'lnlty
.
t he test statlstlc
ln
at t he ra t e0-0.25
n.
k
This indicates that the test statistic lC( and lIV are more unstable than the
test statistic t .
k
The asymptotic distribution of the test statistic t
GN
is
not clear and hence is not included.
3.3
Results for the case 0 > 0.5:
The limiting distributions of
"Stationary Region"
~,
t , lC( and lIV for processes satisfying
k
(2.1) with 0 > 0.5 are summarized in the following theorem.
Theorem 3.3:
(a)
Consider the model (2.1) with p
n -1/2 ~-
and 0 > 0.5.
Then
P -1 ,
---+
, if 0 > 0.625
(c)
(d)
=1
n
-1/2 l
C(-
f1
N
N
3 - 2'
if 0.5 < 0 < 0.75
and
( f)
where NO' N , N and N are independent N(0,1) variables independent of r and
1
2
3
~.
-13-
Note that T and t
k
diverge to negative infinity, whereas (2n)-1/2 la
converges in distribution to a standard normal variable.
probability that l
a
Therefore, the
less than a fixed critical value (negative) converges to
0.5.
That is, in the long run, when 0 > 0.5, the criterion based on Z rejects
a
the hypothesis that the process is nonstationary 50% of the time. The test
criterion based on t , on the other hand, almost always rejects the hypothesis
k
that the process is nonstationary.
Recall that when 0 > 0.5, the process Yt(n)
behaves more like a stationary process than like a nonstationary process and
hence the more often the test statistic rejects the nonstationarity the better
it is.
In this sense, the t-statistic t
statistic.
k
performs better than the Za
However, if you interpret the probability of rejecting
nonstationarity of the model (2.1) as the level of the test for nonstationarity
of the model (1.1), then the statistic Z is doing a "better" job than the
a
statistic t .
k
The order in probability of lIV is greater than that of la'
Also, the
order in probability of Za is greater than that of t , indicating that t is
k
k
more stable than la and ZIV'
Note also that the estimator PIV = 1+n
-1
ZIV
converges in distribution to a Cauchy random variable.
The behavior of the test statistic t
GN
based on the Gauss Newton estimator
is not easy to study under the framework of this paper.
Recall that the
iterated Gauss Newton estimator is the value of the vector (p,e)' that
minimizes
=n
-1 n
E [et(p,e)]
t=l
2
-14-
where
et(p,e)
= Yt
+ (e-p)
t-1 j-1
E e
Yt ..
j=1
-J
Now, ifYt = Yt(n) satisfies (2.1) then, after some algebraic simplification,
we can show that
-&
-1
-1
(e-p)Zt_1 (e) + e t + n Y[(1-e) (1-p)X t _2 -(1-e) (e-p)Zt_2(e) + et - 1 ],
t-1
where Zt-1 (e)
=
e j-1 e t
E
j=1
..
-J
Typically, in the iterative procedures an upper bound on e is set (e.g., e < 0.97
in Said and Dickey (1985».
That is, usually the search is limited to lei < A < 1.
Therefore, if & > 0.5 and lei < A < 1, then with probability one,
Qn (p,e)
uniformly in p and e.
matrix n-
1
2-1
2
-.. Q (p,e) = (1-e) (e-p) + 1,
Q)
The function Q (p,e) is minimized whenever e=p.
Q)
The
E~=1FtFt where Ft is the vector of partial derivatives of
et(p,e) with respect to p and e, converges to a singular matrix.
The asymptotic
singularity of n-1E~=1FtFt makes it difficult to study the properties of
the t-statistic t
GN
.
4.
SUMMARY AND REMARKS
Schwert (1988) cites several examples of economic data that are integrated
first order moving average processes where the moving average parameter is
close to one.
Using an extensive Monte Carlo study, he compares the
-15-
performance of some of the existing test criteria for testing the hypothesis
that the autoregressive parameter is equal to one when the moving average
parameter is close to one.
He observes that the empirical levels of the test
criteria are far from the specified levels, even for large sample sizes.
Similar results are also observed by Hall (1988).
This paper provides the
asymptotic distributions of some of the test statistics as the moving average
parameter approaches one.
We have considered nearly stationary (white noise) processes given by
Yt(n) = Yt - 1 (n) + en - (1-n
-0
Y)e t - 1 .
A nearly stationary process is a
combination of a white noise process and a random walk process.
If 0 is small,
the random walk part is dominant and Yt(n) behaves like a nonstationary
process.
that if 6
If 0 is large, the process behaves like a white noise process.
n = 1-n
-0
.
Y, then 0
= -[~n
n]
-1
[~n(1-6n)
- ltnY].
(Note
For example, with
Y=1, the parameter 6=0.8 corresponds to 0 ranging from 0.5 to 0.26 as n changes
from 25 to 500.
The value 6=0.8 is considered by Schwert (1988) and Hall (1988).)
We have shown that the test statistics t k , t GN , Za and ZIV have the same
asymptotic distributions for 0 < 0 < 0.25 as in the case 0 = 0 (which
corresponds to an invertible integrated moving average process).
We have also
shown that the test statistics are unstable when 0 > 0.25, that is, the
order in probability of the statistics increases with the sample size n.
t-statistic t
k
The
in particular diverges to negative infinity for 0 > 0.25.
Recall that for 0 > 0.5, the process behaves like a white noise process and the
criterion based on t
k
rejects the hypothesis of nonstationarity almost always.
The test criteria based on Z , however, is very unstable and is expected to
a
-16-
reject the hypothesis of nonstationarity only 50% of the time when 0 > 0.5.
No
such conclusions about probability of rejection can be drawn for criteria based
on ZIV and tON'
On the other hand the test statistics Za and ZIV have definite
computational advantages over the test statistics t
that need to be inverted in the computation of t
singular matrices asymptotically.
The test statistic
~
and tON' The matrices
k
and tON are converging to
k
This may cause some computational problems.
is not appropriate for testing the hypothesis of
nonstationarity when the process· is a mixed model.
We included
only to indicate what happens to the test statistic
~
~
in the paper
when the model is
incorrectly specified to be a first order autoregressive model.
The performance of the test statistics t
lags k used in. the computation.
k
and Za depends on the number of
We have taken k
= 0(n 1/ 4 ).
1-e both effect the asymptotic distributions of t and Za'
n
k
example 0
= 0.3
(> 0.25).
For 0
= 0.3,
The orders of k and
Consider for
one may be able to show that the t-
statistic t
has the same asymptotic distribution as in the case of 0=0,
k
prov1'd e d k'1S ta ken to be 0 f or der n0.31 . As we expect, when e is closer to
one, it takes a larger number of lags to approximate an integrated moving average
process by a higher order autoregressive process.
In practice it is, however,
1/4
difficult to decide what the value of 0 is and what multiplier of none
should use to select k.
As pointed out earlier, the composite hypothesis that the process has a
single unit root (nonstationary) includes processes that are close to white
noise.
The present paper studies the asymptotic behavior of the different test
statistics as the models in the null hypothesis approach stationary models.
-17-
Our paper concentrated on test statistics that did not include an intercept
and/or a trend.
For processes satisfying (2.1), with minor modifications, the
results can be extended to test statistics where an intercept is (and a trend
are) estimated.
difficult.
Also, the results for & = 0.25, 0.5, 0.625 and 0.75 are not
The extensions to higher order autoregressive and moving average
processes, however, is not immediate and will be considered elsewhere.
-18-
MATHEMATICAL APPENDIX
Before we present the proofs for theorems in Section 3, we prove a result
and make a few comments regarding the iterated Gauss Newton estimator.
To keep
the proof simple, we use the notation given in Chapter 8 of Fuller (1976) and
Said and Dickey (1985).
Result A.1:
Consider the model
(A.1 )
where p
= 1,
p and
such that p*
~
that in (A.1),
~
eO
= 0,
YO
=0
and {e } is a sequence of iid (0,1) random
t
variables with bounded fourth moments. Let p* and ~* be initial estimators of
1~1<1,
= 0p (n- 3/ 4 ), 1~*1<1
.
1
and ~* - ~
is the same as -e in the model (1.1).)
Gauss Newton estimator
(p,~)'
= 0 p (n- 1/ 4 ).
(Note
Then, the one step
is such that
where
(A. 2)
and
D
n
= d i agona 1
(n, n1/2) ,
(A. 3)
-19-
where r and
~
are defined in Section 2.
t
GN
= (s; c ) -1/2
1
Also, the t-statistic
(p-1)
(A.4)
where
2
se
c
-1
=n
n
E
t=1
- -
[e t (p,J3)]
2
= (1,1)th element of ( En F *F* ') -1
1
t=1 t t
* * *
and F*
t = Ft(p ,13 ) is the 2x1 vector of partial derivatives of e t (p,J3) with
respect to p and 13.
Proof:
-
Note that
= Y t
t-1
E (p + J3)(-J3)j-1 Yt .
-J
j=1
Define,
t
> 0
where eO (p* ,13* ) = 0 and (p* ,13* )' is an initial estimator of (p,J3)'.
Expanding
the function e (p,J3) in a first order Taylor series about the point (p * ,13 * )',
t
we get
-20-
where
V* and Wt* are negative partial derivatives of
t
~
et(p,~)
with respect to p and
evaluated at (p* ,~* )' and St,T t and Kt are second order partial
derivatives of
et(p,~)
with respect to p,
~
and p and
evaluated at a point on the line segment joining
~,
(p,~)'
respectively,
and (p* ,~* )'.
From Said and Dickey (1985), we get for t > 0,
and
where Vo = 0, Wo = 0, So = 0, To = 0 and KO = O.
Note that St
=O.
Let 6 and 6 denote the regression coefficients in the regression of
1
2
et(p* ,~* ) on V*t and W*
t.
[;: ] =
n
E
t=1
n
E
t=1
That is,
V*2
t
**
WtV
t
n
E
t=1
n
E
t=1
-1
**
VtW
t
W*2
t
n
E
t=1
n
E
t=1
V* et(p * ,~* )
t
W* et(p* ,~* )
t
-21-
The one-step Gauss Newton estimator
=
(p,~)'
* *
(p,~)' + (01'
(p,~)'
is given by
°2 )'
Therefore,
n
Dn (p-1,
~_~)
I
=
n
-2 n
V*2
t
I:
t=1
-3/2 n
I:
t=1
where On = diagonal (n,n 1/2 ).
n
-3/2
n
I:
t=1
* *
WtV t
n
-1 n
I:
t=1
-1
* *
VtW t
W*2
t
Note that V* is a function of
t
hence from Said and Dickey (1985) we have
where
and
Then,
~
*
alone and
-22-
and hence,
t-1
*.
*
= E (-~ )J[U t . + (1-p )Yt-1-J']
j=O
-J
After some algebraic manipulations, it can be shown that
*
= (1+~)(1-p
)X t
*
1
*
* t-1
*
"-1
+ (~ + p )(~-~ ) E (_~)J
j=1
et
_.
J
Also, it can be shown that
*
Wt
=
t-2
* j
* *
E (-~) e t - 1- j (p ,~ )
j=O
=
Wt(1,~
*
) +
* -2
(1+~)
(1-p* )X t _
2
*
* -1 *
+ (1-p )(1+~) A t 2
where
t-2
*'-1
* -1 *
*
= E (_~)J ((1+~)· ~ + (~ - ~)j]et-1-J'
j=1
and
= e t-1
*)
Note that
Wt(1,~
+
t-2
*. 1
*
*
(-~ )J- [(~-~ )j - ~ Jet 1 .
j=1
- -J
E
is the same as W considered by Said and Dickey (1985) when
t
the initial estimator for p is taken to be one.
Now, it is easy to show that
-1 n *2
-1 n
* 2
1/2
E [Wt(1,~)] + 0 (n)
nEWt = n
t=1
P
t=1
n
* *
E Vt Wt
-3/2 n
t=1
and
= oP (1)
,
-23-
-1
To complete the proof, we need to show that n
converge to zero in probability.
t-1
= E
j=O
=
n
*
-1/2
E t =1 VtR t and n
n
*
Et =1 Wt Rt
Note that
(-~)j Vt-1-J'(P'~)
t-1
E
j=O
a.e _ _· + a X t 1
J t 1 J
and
=2
t-1
E
j=O
(-~)j Wt-1-J'(P'~)
t-1
=
E
j=O
where a
= 0P (1),
Since p* - 1
b .e t 1 .
J
- -J
j
j
la.1 < M A and Ib.1 < M A for some finite M and 0 < A < 1.
J
= 0p (n- 3/ 4 )
J
and ~*
=
-
~
(1+~)(~
= 0 p (n- 1 / 4 ),
* -~) 2n-1
we get
n
t-1
E X _
E b e _ _
t 1
j t 1 j
j=O
t=1
n X2
+ 2(1+~)(p* -1)(~* -~) a n- 1 E
t=1 t-1
=0(1),
P
+
0
P
(1)
-24and
= n
1/2
= 0p (1)
(p
*
*
-1)(~ -~)
n
* A
E Wt(1,~ )K + 0 (1)
t
t=1
P
•
Therefore, from Theorem 3.1 of Said and Dickey (1985) we get the desired
I
result.
The result (A.1) has several implications.
First, it indicates that the
one-step Gauss Newton estimator p, starting with any initial estimator p *
(not necessarily equal to one), can be used for testing the null hypothesis
that p = 1, provided p* - 1 = 0 (n- 3 / 4 ).
p
Also, since
p-
1 = 0 (n
p
-1
) and
~ - ~ = 0p (n
-
-1/2
estimator.
That is, one can use the iterated Gauss Newton estimator for p to
), we do. not have to restrict to the one-step Gauss Newton
test the hypothesis that p = 1.
The choice of the initial estimator is still
important in the sense we must have p* _ 1
suggest using p* = 1 and
~. = C~:
where c i =
~i+1'
*
~
A2
p
(n -3/ 4 ) .
Said and Dickey (1985)
to be the Durbin estimator
]
1
,
c._
=0
-1 C-1 c.
E
A
. 1
1=
i=1, ... , k-1;
Co
=
~1
'
1 4
+ 1; k = 0(n / ) and
~
=
(~1""'~k)'
are the least squares coefficients in the regression of u = Y -Y - on Y - ,
t
t t 1
t 1
ut - ' ... , ut - k +1 .
1
From the results of Said and Dickey (1985) it can be shown
that
A*
P
=1
* A
+ (1 + ~ )~1
=1+0(n- 1 )
p
-25-
and
~*
So, an alternative initial estimator for p is p .
We will now present simple expressions for the test statistics, considered
in Section 2, that will be useful in deriving the asymptotic distributions.
1.
Consider the T statistic proposed by Dickey and Fuller (1979),
= s1-1( ~
T
y2
L
t-1
t=2
)1/2(~_1)
P
where
s
2
1
=-n -1
n
E (Y
t=2
t
- P Yt - 1 )
2
(A. 5)
and
ut
= Yt
- Yt-1
•
It is easy to see that
(A. 6)
-26-
2.
Consider the regression t-statistic t
k
for testing the coefficient of Yt-1
.
h
k = 0 (n 1/ 4 ) •
. zero ln
. th e regress10n
1S
0 f ut on Yt-1' ut _ 1 ' ... , ut - k + 1 ' were
From (3.1), we have
Ut = -
k-1
E (l-n
_&
~
Y)
Ut_~
+ e
t=l
t
- (l-n
_&
Y)
k
e _ .
t k
(A. 7)
Define, for j=l, ... , k-1, and t>k,
= et _j
- (l-n
k- j-1
=
E
i=O
(l-n
-&
-&
Y)
k-j
et - k
(A. a)
.
1
Y) U
t
.
.
-J-1
Since wt - 1 , ... , wt - k +1 are linear functions of u t - 1 ' ... , u t - k +1' the test
statistic .t
k
is the same as the t-statistic for testing the coefficient of
Y - 1 is zero in the regression of u t on Yt - , w _ , ••• , w - + .
t
1
t 1
t k 1
Note that
from (A.7) and (A. a) we have
+ e
Let
A*
~
t
- (l-n
-&Y) k e _
t k
(A. 9)
denote the regression least squares coefficients in the regression
of ut on Yt - 1 , wt - 1 , ... , wt - k +1 .
Then,
(A.10)
where
-27-
i
E
[ t=k+1
n
t-1
Hk =
h
b
b
k]
,
A
k
k
~
[ t=k+1 Yt-\U t ]
k =
ck
b
n
E Y
=
v - ,
k
t=k+1 t-1 t 1
c
k =
n
E U V _
t t 1
t=k+1
v t - 1 = (w t - 1 '···,
Wt
- k +1 )'
and
n
A =
k
E
t=k+1
Vt_1V~_1
Note that
d
k
-d
k
b
k
-1
A
k
-1
H =
k
-1
-dkAk bk
where
-1
-1
-1
A + A bkbkA k dk
k
k
-28-
Therefore, the t-statistic
-1 1/2 [
t k = sk dk
n
_ b' A-1
]
E Yt-1 Ut
k k ck
'
t=k+1
(A. 11 )
where
2
sk = (n-2k)
[nE
-1
t=k+1
2
ut -
-1
C~Ak Ck
- dk
[
n
E Y
t=k+1
.t
t-~
3.
The test statistic t
4.
It is easy to show that the test statistic Z given in (2.8) can also be
ex
GN
is as given in (2.7).
written as
Zex. = (n
5.
-1 n
2
E Yt -1 )
t=1
-1 [ n
n
k
E Yt -1 u t - E
E u t Ut _ .
t=2 .
j=1 t=j+1
J
]
(A.12)
Finally, the test statistic ZIV given in (2.10) can be written as
[ n
-1 n
ZIV = n ( E Yt - 2Yt - 1 ) (E Yt - 2Yt ) - 1]
t=2
t=2
n
n Y- Y - ) 1
= n[ ( E
E Yt-2Ut
t 2 t 1
t=2
t=2
].
(A.13)
We will now prove a Lemma that will be used in deriving the asymptotic
distributions of the test statistics.
Lemma A.1:
Consider the vector a =
(ex"
••• I
Let 1 = (1, ... ,1)' is a (k-l) x 1 vector with all coordinates equal to one.
1 4
Then, for k = o(n / ), as n tends to infinity,
-29-
(i )
(1-n
-0 k
Y)
~
0 if 0 < 0 < 0.25 ,
(iii) n-ocx'cx--+ (2y)-1 if 0 < 0 < 0.25 ,
(v)
n
(vi) k
-0
-1
1'cx--+Y
1'cx
=1
-1
if 0 < 0 < 0.25 ,
+ O(n
-0
k)
if 0 > 0.25 ,
and
if 0 < 0 < 0.25
(viii). (1 - cx) I (1 - cx)
if 0 > 0.25
Proof:
-
Note that
and
Therefore, for 0 < 0 < 0.25,
(1_n- oy)k converges to zero (exponentially).
Result (ii) follows from the first order Taylor series expansion of (l_n- oy)k.
Now,
-30-
k-1
a'a = E
.
(1_n-Oy)21
i =1
-1
= [1-(1-n-Oy)2]
= [2Y
(1_n-Oy)2[1_(1_n-Oy)2(k-1)]
- n-Oy2] -1 (1_n-Oy)2 nO[1_(1_nOy)2(k-1)]
Therefore, the results (iii) and (iv) follow from the results (i) and (ii).
1 'a
=
k-1
Also,
.
E (l-n -Oy)'
i=1
= y-l no(1-n-oY)[1-(1-n-oy)k-1]
and the results (v) and (vi) follow from the results (iii) - (iv).
(1 - a) , (1 - a)
=k
- 21 'a + a 'a
and the result (vii)"follows from the results (iii) - (vi).
Notation:
Finally,
We use the Euclidean norm "x"
= (x'x)1/2,
I
of a column vector x to
define a matrix norm "B", when
IIBII
= sup{"Bxll
"x"
< 1) •
Recall that "B"2 is bounded by the sum of squares of the elements of B and that
"BII is bounded by the largest modulus of the eigenvalues of B.
-31-
We are now ready to derive the asymptotic distribution of the test
statistics.
To save space, we will suppress n and write Y for Yt(n).
t
Proof of Theorem 3.1:
Recall that
-&
Yt = e t + n
~
Xt - 1
t
X
t
= I: e.
i =1 '
and we assume that 0 < & < 0.25 and 0 <
(a)
~
< 2.
Note that
n 2
I: Yt - 1
t=2
=
n-1 2
I: e
t
t=l
+
n
-20 2 n-1
~
I:
t=1
X
2
_& n-1
2n
~
I: x _ e
'
_
+
t 1
t 1 t
t=1
(A.14)
and wi th u
t
-(l-n -& ~)
n
2
I: e t - 1 +
n
-20_2
r
t=2
+ (1-n
_&
~)
n
I:
t=2
Therefore,
-1
n
and for 0 < & < 0.25,
n
-2+2&
n
2
E
I: Yt-l~
t=2
r
2
L
-6
e t _1e t + n
~
X le
n- n
(A.15)
-32-
Also, note that
-1 n 2
-1 n 2
-0 2 n 2
-0
-1 n
I: e + (1-n Y)
I: e - - 2(1-n Y)n
I: e _ e
n
I: u = n
t 1 t
t
t
1
t
t=2
t=2
t=2
t=2
P
--2.
(A.16)
Therefore, from (A.6),
n
n
= (S2 n- 2+20 I: y2 )-1/2 n -l I: y
u
1
t=2 t-l
t=2 t-l t
Consider the t-statistic t
(b)
-(1-n
-0
Y)·+
given in (A.11).
k
n
-20_ 2
~
n
-1
Note that, from (A.15),
n
I: X _ e + 0 (n
t=2 t 1 t
P
-1/2
).
(A.17)
Also, for i, j = 1, ... , k-1,
-1
n
n
-1
n
-0 k-i
-0 k- j
I: wt _ 1.wt _. = n
I: [e t 1. - (l-n Y)
et-k][e t - ·- (l-n Y)
et - k ]
J
J
t=k+1t=k+1
where 0 .. is one if i=j and zero otherwise.
Therefore,
1J
Q
= II n -1 Ak -
I -
a a' II
where A is defined following (A.10) and a is defined in Lemma A.1.
k
to show that the eigenvalues of
(I + aa')
-1
=I
(I.~
a')
-1
- (1 + a'a)
are (1+«'a)
-1
aa' .
-1
and one.
It is easy
Also,
(A.18)
-33-
Therefore,
= II
P
,
(I +<1<1 )-1 11 < 1
and
QP
S (q + p)
1 4
Since Q = 0p (n- / ) and p
•
= 0(1),
we get
= 0 p (n- 1 / 4 )
q
(A.19)
and
(A.20)
Consider now, for j=1, ... , k-1,
n
-1
n
E
t=k+1
wt_.u t = n
J
n
-1
E
t=k+1
= -(1-n
-&
(e
Y)n
. - (1-n
t- J
-1
-&
Y)
k- j
et_k](e t - (1-n
_&
Y)e t - 1 ]
n
1/2
E e _ .e _ + 0 (n)
t=k+1 t J t 1
P
Therefore,
lin
-1
c k + (1-n
-&
Y) E11I
= 0 p (k 1 / 2 n- 1 / 2 )
(A. 21 )
-34-
where E
1
=
(1,0, ... , 0)1 and c
is defined following (A.10).
k
Also, for
j=l, •.. ,k-l,
n
-1
n
E
w .Y
t=k+l t- J t-
1 =
n- t=k+1
~ [e .
t-J
1
+ n
-0
-1 [
Yn
n
E X - e _. - (1-n
t=k+l t 2 t J
-0
Y)
k- j
Note that
-1
n
n
E X
~
2,
-1
Therefore,
-1
n
-1
- n
t=k+1 t-2 t-k -
and for j
n
e
n
E
t=k+l
wt - 1 Yt - 1
n
E X
e
t=k+1 t-k-l t-k
-1
n
+ nEe
2
t=k+l t-k
k-l
+
E
R=2
n
]
E Xt - 2e t - k
t=k+l
-35-
and for j
~
2,
-1
n
Therefore,
lin
-1
bk
(1-n
-0
Y)E
1
(A.22)
From (A.21) and (A.22), we have
lin -1 c
k
II = o
P
(1)
and
if 0 < 0 5: 1/8
if 0 > 1/8
Therefore, for 0 < 0 < 0.25,
(A.23)
Also,
n
-1+o 'A.- 1
b k-l< c
k
= -n o
n x _ e
(1-n -0Y) 2 - Y[ 1+n -1 E
t=2 t 1 t
J+
(1) ,
0
P
-36-
and hence
- 1 En X _ e J+0 (1)
= -n o(1-n -0Y) + n0 (1-n -0Y) 2 + Y(1+n
P
t=2 t 1 t
= Yn
-1 n
E X _ e + 0 (1) •
t=2 t 1 t
P
(A.24)
Recall that
sk2 = ( n- 2k)-1
= n -1
p
1c ] 2
En u2 - n -1 c'k A-k1c - n -1 dk [ E
n Yt-1 Ut - b 'A.t
k
k-K k
t=k+1
t=k+1
n u 2 - n-1 e'(I
+ ,..".,,)-1
E
1
~
e 1 + 0p(1)
t
t=1
2-1 = 1.
--+
(A.25)
From (A.11), (A.23), (A.24) and (A.25), we get
(c)
Consider the test statistic t
GN
given in section 2.
Let p * and J3 * be
(=n -0 Y-1) such that p* - 1 = 0p (n -1 ), 113* I <1
n
= 0 (n-0.5-0.50), for 0 < 0 < 0.25. (It can be shown that the
initial estimators of p (=1) and J3
and J3* - J3n
estimator J3
p
r
considered in Said and Dickey (1985) satisfies this condition.)
Then,
n
-0
*
(1+13) -
P
-1
Y
and
*
( 1 +13)
-1 ( 1 +J3 ) - P
n
1 .
-37-
Following the arguments given in Result A.1, it can be shown that
-0
= n \-1
+ ft
and
where
*
Bt = (1-p P'
*
-1 0
* -1 0 t-2
* j-1 a.**
n X _ 2 + (1-p)'>' n
I: e 1 . (-/3)
t
t
j=1
- -J
J
- (/3 -/3n)
t-2
I:
* . 1
j[(-/3 )J-
A* j-1
+ (-/3)
j=1
]e - 1t
j
a.** = (/3 * - /3 )j + '>' -1 n 013 *
n
J
is a point between /3 * and /3 •
n
After some tedious algebra it can be
shown that
-2 n
n
*2
I:
t=1
n
-1-0
Vt
n
I:
t=1
*2
= (2'>') -1 +
W
t
n
0
P
(1) ,
n-1.5-0.50 I: w*v* = 0 (1) ,
t=1 t t
P
-38-
and
n
-0.5-0.50
E
W*tet = 0 (1) .
n
P
t=1
Now, from the arguments used in the proof of Result A.1, it follows that
(d)
To derive the asymptotic distribution "of Z
a:
n
N = E Yt-1Ut
t=2
-
n
= E Yt - 1Ut
t=2
-
=
k
n
E
E
j=l t=j+l
UtU t _.
J
k
t-l
E u
ut .
-J
j=1
t=2
t
in (A.12), consider
E
-
n
E
k
u
E u _"
t=k+l t j=1 t J
n
E Yt - k- 1u t
t=2
After some routine algebra, it follows that
N=
+ n
-20.2
,
Therefore, for 0 < 0 < 0.25,
(A.26)
-39-
and
(e)
To find the distribution of ZIV given in (A.13), consider
(A.27)
and
(A.28)
Therefore, from (A.13), (A.27) and (A.28), we get for 0 < 0 < 0.25,
= [n- 2+20
n
1 -1+20 n
E Y Y ]- [n
E Y- U
t=3 t-2 t-1
t=3 t 2 t
]
The following lemma is used to prove Theorems 3.2 and 3.3.
Lemma A.2:
Let {e } be a sequence of iid (0,1) random variables with bounded
t
fourth moments.
Consider the random variables,
=n
-1/2 [nr]
( E e t'
t=1
n
*
E E ' e )'
t=1 t
where
E*t = (e t - k - 1 , e t - k , e t _2 , e t - 1 )',
t
-40-
[nr] denotes the integer part of nr, 0 < r < 1 and k = 0(n 1/ 4 ).
Then,
(Sn(r), Nn')'
(W(r)
•
N
'3'
N
2'
N
l'
N)I
0
'
where W(.) is a standard Brownian motion, NO' N , N and N are independent
1
2
3
N(0,1) variables independent of W(.).
Proof:
Let 7 t denote the a-field generated by {es:s
(E nt =1 Et* e , 7 t ) is a martingale sequence.
t
n
n
-1
E
t=k+1
E
* *
t
E
t
1P
1
~
t}.
Note that
Also,
4
where 1 denotes an identity matrix of size 4.
4
Since et's are assumed to have
bounded fourth moments, the result follows from Theorem 2.2 of Chan and Wei
(1988) and Theorem 3.3 of Helland (1982).
I
We will now present a proof of Theorem 3.2.
Proof of Theorem 3.2:
Recall that the process of Y satisfies
t
Y = e + n -&Y X - ,
t
t
t 1
0.25 < & < 0.5 , 0 < Y < 2.
t
where Xt = E.,= 1e ,..
(a)
From (A.14), (A.15) and (A.16) we get, for 0.25 <
& <
0.5,
-41and
n
-1 n
E u
t=1
2
-
t
P
2 •
Therefore, from (A.6) we have
(b)
Note that, from (A.17), (A.21) and (A.22) we have
= -1
+ E, (n
.
-1
'\)
-1
E
1
+ 0 (k
p
-1
)
•
(A.29)
We will now show that
k(-1 +
E,
(n
-1
Ak )
Notice that Ei (n-1 ~1 )E
1
P
E1-J--
-1
-1.
is the first element of (n -1 Ak ) -1 which is also
1 2
the inverse of the residual sum of squares in the regression of n- / w _ on
t 1
-1/2
-1/2
-& k-i
n
wt - 2 ' ... , n
wt - k+1 ' where wt - i = e t - i - (1-n Y)
et - k
Consider
the model
n
-1/2
wt - 1
=
k-1
E
ain
-1/2
i=2
wt - i + error.
(A.3D)
The model (A.3D) is equivalent to
n
-1/2
et - 1
=
k
E a.n
.i=2 -1
-1/2
e
t
. +
error,
(A.31)
-1
where
(A.32)
-42-
Therefore, the residual sum of squares of the regression (A.3D) is the same as
the residual sum of squares of the model (A.31) where the coefficients satisfy
the condition (A.32).
Let
n
E
t=k+1
Vt_1V~_1
and
Then, the condition (A.32) is a'a = (1_n- oy)k where a is defined in Lemma A.1,
and the residual sum of squares of the regression (A.3D) is
1
1
(E '(n- A. )- E ]1
1
-1<
1 = n
-1 E
n e2
- a'R a
t=k+1 t-1
k
A_1
+ (a'R
From Berk (1974) we get,
a'a -
p
0,
and
a'R a
k
= 0p (n- 1k)
k
a)
-1
A
0
k 2
[a'a - (1-n- Y)]
•
(A.33)
-43-
Now, from (A.33) we have
p
-1
(A.34)
.
From (A.ll), (A.23), (A.25), (A.29) and (A.34), we get
(c)
From (A.26) and (A.14) we have, for 0.25 < & < 0.5,
and
Therefore, from Lemma A.2 and (A.12), we have
(d)
From (A.27) and (A.28) we have for 0.25 < & < 0.5,
n
-2+2& n
2 2 n 2
E Yt-2 Yt-1 = Y n- E X
+ 0 (1)
P
t=3
t=3 t-1
and
-1/2 n
n
E Yt - 2Ut
t=3
= n-1/2
n
E e _ e
t 2 t
t=3
-1/2 n
- n
E et_,e
t=3
t
(1)
+ 0
P
.
-44Therefore, from Lemma A.2 and (A.13) we get
I
Finally, we present a proof for Theorem 3.3
Proof of Theorem 3.3:
Y
= et
=
E.1= 1e.1
t
where \
(a)
+ n
Here we assume that
-0
Y Xt - 1 '
0 > 0.5,
a <Y< 2
t
For 0 > 0.5, note that
-1 n
n
n
2
E Yt - 1
t=2
-1 n
E u
2
t=2 t
-
-
P
P
2
1 ,
I
and
2
-1 n 2
-1 n 2
-1 -1 n
2
s1 = n t E
=2 Ut - [n
E Yt - 1 ] [n
E Yt - 1 Ut ]
t=2
t=2
P
---
1.
Therefore, from (A.6), we get
n-1/2
(b)
A
P
't' --+
-1 .
The results (A.29) and (A.34) hold also for 0 > 0.5 and
= _k- 1
+ 0 (k- 1 )
'p
-45-
Also,
For 0.5 < 0 < 0.625,
n-2+20
n
E
y2
t=k+1 t-1
n
-2+2Ob 'A-'b
k k
k
and
1.
Therefore, for 0.5 < 0 < 0.625,
(c)
For 0 > 0.625,
p
2 P
sk - 1 ,
1,
-46and
Therefore, for 0 > 0.625,
(d)
Note that for 0 > 0.5, from (A.26) we have
n
-1/2
N= n
-1/2
n
-1/2 n
E e t - k - 1e t - n
E et_ket + 0 (1)
t=k+1
t=k+1
P
and
n
-1
n
E
-1
y2
t=k+1 t-1
n
2
+
t=k+1 t-1
= nEe
(1)
0
P
Therefore, from (A.12) and Lemma A.2, we get
n
(e)
-1/2 Z
~
ex-
N
N
3 -
2'
For 0.5 < 0 < 0.75, from the arguments used in the proof of Theorem 3.2,
we get
-2o+0.5 Z
(f)
B
IV-
n
[2 ]-1(N
YT
1-
N)
0
.
For 0 > 0.75,
-1
n
ZIV + 1
=[
~
y
y
t=3 t-2 t-1
-1
N1 No·
] -1 [
~
Y
Y
t=3 t-2 t
I
J
-47ACKNOWLEDGEMENTS
I am grateful to David Dickey for his comments and suggestions.
also go to the National Science Foundation for research support.
My thanks
Finally, my
thanks go to Janice Gaddy for her skill and effort in typing this manuscript.
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