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Working Paper 9014
COINTEGRATION AND TRANSFORMED SERIES
by Jeffrey J. Hallman
Jeffrey J. Hallman is an economist at
the Federal Reserve Bank of Cleveland.
Working papers of the Federal Reserve
Bank of Cleveland are preliminary
materials circulated to stimulate
discussion and critical comment. The
views stated herein are those of the
author and not necessarily those of
the Federal Reserve Bank of Cleveland
or of the Board of Governors of the
Federal Reserve System.
December 1990
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1
I.
Introduction
A large and growing literature is concerned with the theory,
estimation,
and
applications
of
cointegrating
vectors
and
associated error correction models. A cointegrated system is a set
of time series that individually follow difference-stationary
linear processes, but one or more linear combinations of the series
do not require differencing to appear stationary. The stationary
linear combinations indicate stable long-run relationships. Engle
and
Granger
(1987)
demonstrate
the
correspondence
between
cointegrated time series and error correction models:
generating
processes
correction
for
cointegrated
systems
have
error
representations, and error correction models generate cointegrated
series.
Nearly all of the work in the unit root literature thus far is
applicable only to series generated by a linear process.
exceptions are two papers by Granger and Hallman (1988, 1990)
The
.
The
first of these considers properties of nonlinearly transformed
integrated series and the effect of such transformations on unit
root tests. The second introduces the concept of "attractor sets,
a nonlinear generalization of cointegration. Roughly speaking, if
x, is an n-dimensional time series with all components having long
memory (defined below), then a subset A of Rn is an attractor set
if z,,
the Euclidean distance from x, to A, is a short-memory
process with bounded variance.
Linear cointegration is a special
case in which A is a hyperplane, and the components { x i , ) of x, are
not only long memory but difference stationary as well.
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2
This paper can be regarded as falling between the studies
described
above,
focusing
after
on
series
they
are
that
are
individually
(linearly)
cointegrated
only
nonlinearly
transformed.
Such series may be thought of as having an attractor
that is the kernel of an additively separable function of
X, = (
A =
xlttx2,,
...
);
that is,
{ x : f(x) = 0 )
where
but this is not always true.
Nonlinear cointegration is more
general than the notion of an attractor set and may be more useful
to economists as well. The relationship between attractor sets and
nonlinear cointegration is explored in section 11.
If two series are cointegrated and the second series is also
cointegrated with a third, then it is well known that the first and
third series are also cointegrated.
Granger and Hallman (1988)
show that an integrated series is not cointegrated with a nonaffine
transformation of itself.
From this it follows that if f(x,) and
g(y,) are cointegrated, f (x,) and h(y,) are not, making it important
to get the transformations right.
By allowing for nonparametric
transformation of the series as part of the estimation procedure,
the two algorithms outlined in section I11 increase the odds of
finding long-run relationships if they do exist.
Section
IV
discusses
testing
for
cointegration
among
transformed series and is followed by some illustrative examples in
the fifth section.
Section VI concludes.
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3
Attractor Sets and Nonlinear Cointegration
11.
Start with some definitions from Granger and Hallman .(1990).
Let the information set I, be defined as I, =
2,
. ..
}
{
x,-~,
Qt-j: j
= 0,
1,
, where Q, is a vector of other explanatory variables.
Then the series x, is said to be short memory in distribution (SMD)
with respect to I, if
as h
- - for all appropriate sets
A
and B.
If equation (1) does
not hold, x, is called long memory in distribution (LMD)
.
Denoting
=
F, where
the conditional expectation as
x, is said to be short memory in mean (SMM) if limf,,,
h--
F is a random variable with a distribution that does not depend on
I,.
If f,,, depends on I, for all h, then x, is long memory in mean
(m)
The univariate series x, has a point attractor m if x, is SMM
with limf,,, = m and the forecast error e,,, = (x,+, - m) has bounded
h--
variance as h
-
a.
Similarly, the n-dimensional series x, is said
to have an attractor A
E
R n if z , ,
the signed Euclidean distance
from x, to the nearest point in A, is SMM and has finite variance.
It is obvious that x, has the point attractor m = (m,,, m,,,
... ,
mnt
)
if its components {xi,)are each SMM with mean mi,. Two interesting
cases analogous to cointegration arise when the components {xi,)of
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4
x, are LMM and either (i) x, has an attractor or (ii) a nontrivial
function f
:
Rn
+
RI
exists such that f (x,) is SMM.
This second
case will be called nonlinear cointegration, and the function f
will be referred to as a cointegrating function.
If x, is LMM with an attractor, it is also nonlinearly
cointegrated,
with
the
Euclidean
distance
cointegrating function (there may be others).
function
as
one
However, the notion
of an attractor may be overly restrictive, since it is possible for
series
to
be
nonlinearly
cointegrated
in
an
economically
interesting way without having an attractor. To see in general how
this can happen, suppose f (x,) is a cointegrating function with
mean zero and kernel
A = {
x : f(x)
A;
that is,
= 0 ).
<
is the closest point in A to x,, then by the Mean Value
If
Theorem there exists a real number q , E [0,11 and a point
x,*=
rltxt +
A
(l-tlt)xt
such that
f(x,) = f(xp)+ Vf (x:) '(x,- xp)
=
Vf (x;)*(x,- xp)
I
since f(xp)= 0. Let 8, be the angle between Vf (xi) and (x,- xp) ,
and let z, be the signed Euclidean distance from x, to xp.
implying that
Then
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If the denominator of equation (2) is bounded away from zero (that
is, 3 6 > 0 s.t. Icos0,l ((Vf
(x,')
(1 > 6 )
,
then the finite variance
property of f (x,) will carry through to
2,.
Given the bound, lz,(
may be thought of as the product of two series, at least one of
which (f[x,]) is SMM.
Granger and Hallman (1988) show that for
linear series, the product of an I(0) series with either another
I(0) series or an I(1) series is sMM.'
This suggests that in many
cases the right side of equation (2) will also be SMM, so that the
kernel of f will be an attractor.
However, if the denominator of
equation (2) tends to zero as t gets large, the finite variance
property for z, required by the definition of an attractor may not
hold.
Bounding Icose,( away from zero seems reasonable enough, since
it is zero only if Vf evaluated at x,* is perpendicular to Vf
evaluated at the (nearby) point x:.
For economically interesting
functions, this seems unlikely to happen.
For example, if f is
additively separable of the form
'~ctuall~,
they show that the product of an I(0) and a random
walk is SMM, but since an 1(1) series can always be written as the
sum of a pure random walk and an I(0) series, the result follows.
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f (x) =
@,
(x,) +
$2 ( ~ 2 )+
+
@Xn (Xn)
1
then the gradient of f at xi can become perpendicular to the
: only if some of the slopes of the {Gi
gradient at x
Requiring monotonicity of the
{@i)
)
change sign.
is enough to prevent this.
However, going further and also bounding IIVf(x:) 11 away from zero is
For example, the log of the U.S. M2 money
quite restrictive.
supply follows an integrated process with positive drift and is
cointegrated with the log of nominal GNP. The cointegrating vector
is (1,-l), so that the log of M2 velocity is stationary around its
mean of 0.50077 (= log[l. 651). But while there is an attractor for
the logs of money and income, there is no attractor for their
levels.
Define f by
f (Y,,M,)
=
log (Y,) - log(M,)
-
.50077.
The candidate attractor set is the kernel of f in Y-M space:
A = {
(M,Y) : Y
-
1.65M= 0
).
The linear trend in log(M,) translates into an exponential trend in
the levels of M,, Y,, so that the gradient
is asymptotically driven to (0,O). If log velocity has a constant
variance, the variance of the Euclidean distance from (M,,Y,) to the
line Y = 1.65M grows like eZt.
A
is not an attractor for M, and Y,,
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7
despite the fact that they are nonlinearly cointegrated.
The point of this example is that the existence of an
attractor between two or more series is not robust even to
invertible transformations of the individual series.
true for nonlinear cointegration.
This is not
If x, and y, are nonlinearly
cointegrated, then so too are g(x,) and h(y,), if g and h can be
inverted.
111.
Estimation
Cointegrating transformations are
not
generally
unique.
Granger and Hallman (1990) show that if x,, y, are cointegrated,
g(x,) and g(y,) are also cointegrated if either (i) g is homogenous
or (ii) the series are scaled so that the cointegrating vector is
Absent
(11-1)
further
structure,
estimating
a
pair
of
cointegrating transformations for x,, y, is not a well-defined
optimization problem.
An optimization problem that can be solved nonparametrically
is finding the transformations @ ( . ) , 8(- ) that maximize the sample
correlation of
between
@I (x,)
and 8 (y,)
.
Since the asymptotic correlation
cointegrated series is one, one can hope that the
correlation-maximizing transformations will also cointegrate.
If
the llequilibriumerror" 8 (y,)- @I(x,) is thought to be stationary as
well as SMM, the maximization can be carried out subject to the
restriction that the variance of
constant.
the estimated residual
is
There is no guarantee that either of these approaches
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will always discover a pair of cointegrating transformations if
they exist, but applying the methods at least provides a start.
Alternating Conditional Expectation (ACE) is an algorithm
proposed by Breiman and Friedman (1985) to find transformations
(8,@l,@2,... )
for a set of variables
(y,x,,x.
x@i .
x )
that
n
maximize the correlation between 8(y) and
(xi)
This is
i =L
equivalent to maximizing the R~ from a regression of 8(y) on
(xl),
...,@,, (xn), or minimizing
The steps in the ACE algorithm are as follows:
(ii) Iterate until e2
. .,@,)
(a) Iterate until e2(8 ,el
fails to decrease:
. . ,en) fails to decrease:
F o r k = 1 to n : Set
i *k
End inner iteration loop;
I
@,+E (8(y) -x@i(xi)) I x,;
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Set
End outer iteration loop.
Upon
0
completion
of
the
algorithm,
...,cpn minimize equation
@
,,
Tibshirani's
(3)
the
transformations
.
(1988) additivity and variance stabilization
(AVAS) algorithm is a modification of ACE that chooses 0(y) so as
x@i
n
to achieve a stable variance for the residual e,
=
0 (y,) -
.
(xi,)
i =l
At each iteration the variance function
V(U) =
I
var 0(y) (
C@i
(xi)= U
i=l
1
is used to compute the variance-stabilizing transformation
0(y) for the current iteration is then computed as 0(y) +g[0(y)
]
fromthe previous iteration, standardized to mean zero and variance
one.
For the examples in section V with trending economic times
series, AVAS yields more sensible transformations than does ACE.
Having estimated the transformations
{@lr@21..
.
, it may be
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10
desirable
to
obtain
transformation.
fitted values
for
y,
rather
than
its
This can be done either by finding
or by simply inverting 0(y) if it is monotone.
Of course, the conditional expectations appearing in equations
(3) and
(4) are not usually known and have to be estimated.
Breiman and Friedman suggest using data smooths in their place.
Any one of several scatterplot smoothers can be used, including
splines, nearest neighbor, and regression smooths.
(See Silverman
[I9851 and his discussants for a survey on the use of splines for
scatterplot
smoothing, and
Cleveland
procedure. )
In AVAS, the variance function v(u) is obtained by
[I9791
for
his
lowess
smoothing the logs of the squared residuals {r,) against the fitted
values
I
C mi (xi,)
and exponentiating.
See Tibshirani (1988) for
details.
In the ACE routines used for this paper, both fixed and
variable window regression smooths are employed.
smooth of size k computes E (y
(i)
I
A fixed window
x ) as follows:
Sort the observations by x value.
(ii)
Define the window Wn as the set of all observations
{xjtyj)
such that
k + 1.
I j -n 1
I k t so that the minimum window size is
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11
(iii)E (y,
I
x) is the fitted value of y, from a linear
regression of y on a constant and x, using only the
observations in the window W,.
(iv) Repeat steps (ii) and (iii) for each individual
observation y, in y.
(v)
For technical reasons detailed in Breiman and Friedman,
the data smooths must always have a zero mean, so the sample
mean of the computed E (y
I
x) is subtracted away before the
observations are sorted back into their original order.
If k
+
1 = T I the sample size, the smooth is just the linear
regression y
=
fitted values
{
a + px and the returned values are the demeaned
-
.
At the other extreme, k
minus its mean, a perfect fit.
= 0
will return y
In between, larger values of k
trade more smoothness for less ability to trace discontinuities and
sharp changes in the slope of y
I
x.
The effect of reducing the
window size is similar to what happens in a linear regression as
more variables are allowed to enter.
The smoother used in Breiman and Friedman's ACE implementation
is the variable window llsupersmootherll
of Friedman and Stuetzle
(1982).
It differs from the fixed window smoother by making
several passes with different window sizes and then choosing one of
these for each observation based on a local cross-validation
measure.
When there is substantial autocorrelation among the
prediction errors of the sorted data, the supersmoother tends to
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12
choose window sizes that are too small, so that a plot of the
smoothed data may still appear somewhat jagged. Experience so far
indicates that this effect is mitigated by a high signal-to-noise
ratio, as when the series are highly correlated after very smooth
transformations. Nonlinear cointegration is expected to be such a
case, and the transformations of economic series found by the
supersmoother in section V appear acceptably smooth. Nonetheless,
both fixed and variable window smooths are employed in the ACE
regressions given in sections IV and V to explore the effects of
changing window sizes. Only a variable window smooth was available
in the AVAS implementation.
Breiman and Friedman prove that for a stationary, ergodic
process, ACE converges to the optimal transformations if the
smooths used are (i) uniformly bounded as T
(iii) mean-squared consistent.
-,
a,
(ii) linear, and
Marhoul and Owen
(1984) show
regression smooths to be mean-squared consistent under conditions
that are not satisfied by integrated time series.
No one has yet
derived conditions under which ACE and AVAS are asymptotically
guaranteed to find cointegrating transformations if they exist.
The approach taken here is to use the algorithms to find candidate
transformations and then test for cointegration as outlined in the
next section.
IV.
Testing
If x, and y, are LMM series with cointegrating transformations
f (x,) and g(y,), then z, = g(y,)
-
f (x,) is SMM. Evidence that f (x,)
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and g(y,) are LMM while z, is not is one way to test for nonlinear
cointegration.
Granger and Hallman (1988) propose using both the
Augmented Dickey-Fuller (ADF) test and a rank version of the ADF
called RADF to test the LMM property in a univariate series.
The ADF statistic for testing the unit root hypothesis is the
t-statistic for a in the regression
If p = 0 , no lags of Az, appear in equation (5).
The resulting
statistic is then referred to as the Dickey-Fuller (DF) statistic.
A one-sided test of the hypothesis of a unit root in z, is rejected
if the statistic falls below a critical value.
If z, has a nonzero
mean, either it is subtracted off before performing the test, or a
constant is included in the regression.
If z, is a residual from
ACE or from a regression including a constant term, it has mean
zero by construction.
To construct the RADF statistic, let r,
=
rank(z,); that is,
r, is one if z, is the largest of the z ) , or two if z, is the
second largest of the {z,), and so on. Replace the {z,) in equation
(5) by their ranks and then compute the RADF as the t-statistic for
a just as before.
By construction, the RADF and RDF
(rank
counterpart of the DF) statistics are invariant to monotone
transformations of z,.
Granger and Hallman (1988) show that this
is a considerable advantage in that the usual DF and ADF tests
perform
badly
when
z,
is a
nonlinear transformation of
an
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14
integrated series with a linear generating process.
Use of the ADF as a test for linear cointegration was first
suggested by Engle and Granger (1987), and its distribution has
been studied by Phillips (1987), Engle and Yoo (1986), and Yoo
(1987).
test.
Engle and Yoo provide tables of critical values for the
These depend on both the number of observations in the
sample and the number of parameters estimated in the cointegrating
regression.* This presents a problem because ACE and AVAS do not
estimate parameters.
However, shrinking window sizes in ACE is
much like allowing for more parameters in a regression.
What is
needed is an indication of the effects of changing window sizes on
the distribution of ADF and RADF statistics constructed from ACE
and AVAS residuals.
A simple Monte Carlo experiment was conducted using 300
repetitions of the following:
(i) Generate u and e as vectors of 100 i.i.d. N(0,l) random
variables,
t
(ii) Form summations xt = x u j , et
j=I
t
=
x e j I and
j=I
(iii) Form yt by
(a) Yt
= Etr
1
(h) yt = ?xt + E ~ and
,
See table 3 (panel b) for percentiles of the RADF as a test
for linear cointegration.
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(c) y,
=
3x, +
E,.
If the series were stationary, (a), (b), and (c) would
correspond to R' values of 0, 0.1, and 0.9, respectively.
In
fact, y, and x, are correlated random walks that are not
cointegrated.
(iv) The ACE algorithm was applied to the series with various
fixed window sizes, as were both ACE and AVAS algorithms using
the variable window size smoother.
All transformations were
restricted to be monotone. After forming the residual series
z, for each case, the ADF and RADF statistics were computed
with four lags of Az, appearing on the right side of equation
Results of the simulations are summarized in tables 1 and 2,
which show the percentiles of the ADF and RADF distributions
generated by the experiment. As in Engle and Yoo, the minus signs
are omitted for simplicity.
For comparison, table 3 shows the
distributions of the ADF and RADF statistics using residuals from
ordinary least squares (OLS) regressions of a pure random walk on
a constant and one, two, three, or four independent random walks.
Again, four lags of
Az, were used in the ADF regression.
This
table is based on 5,000 replications of each test.
Several patterns are evident in the tables.
From the fixed
window entries, it is apparent that both the ADF
and RADF
distributions shift to the right with increasing window size and
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16
increasing p .
The RADF results for the variable window ACE and
AVAS appear stable across the three
for AVAS.
P values, as do the ADF results
The ADF distribution for the variable window ACE shifts
right as p increases.
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Table 1
Method
Window
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
AVAS
9
14
19
24
29
34
39
44
49
Variable
Variable
ADF Percentiles
5%
10%
20%
50%
80%
90%
95%
1.97
1.67
1.53
1.33
1.18
0.91
0.89
0.92
0.60
1.01
0.84
2.17
1.91
1.74
1.63
1.49
1.37
1.35
1.27
1.19
1.36
1.35
2.40
2.26
2.11
1.99
1.90
1.77
1.74
1.72
1.63
1.81
1.76
3.17
2.96
2.81
2.66
2.55
2.53
2.48
2.42
2.36
2.55
2.45
3.66
3.58
3.49
3.33
3.37
3.30
3.18
3.17
3.07
3.43
3.16
4.08
3.98
3.79
3.77
3.72
3.67
3.53
3.52
3.49
3.95
3.47
4.35
4.32
4.20
4.15
4.11
3.88
3.88
3.77
3.72
4.45
3.77
3.04
2.85
2.73
2.59
2.54
2.46
2.38
2.36
2.26
2.67
2.46
3.73
3.50
3.38
3.25
3.20
3.14
3.08
3.02
3.02
3.51
3.11
4.09
3.83
3.75
3.62
3.47
3.45
3.40
3.39
3.31
3.92
3.47
4.24
4.16
4.06
3.99
3.85
3.71
3.62
3.60
3.54
4.45
3.82
(b) ,3
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
AVAS
9
14
19
24
29
34
39
44
49
Variable
Variable
1.82
1.59
1.42
1.35
1.24
0.97
1.05
0.88
0.80
1.18
0.91
2.11
1.97
1.78
1.67
1.58
1.42
1.37
1.29
1.28
1.51
1.31
=
2.38
2.20
2.11
1.95
1.86
1.85
1.76
1.69
1.66
1.90
1.75
(c) 8
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
AVAS
OLS
9
14
19
24
29
34
39
44
49
Variable
Variable
0.333
=
3
1.26
1.12
1.00
1.02
0.78
0.81
0.75
0.66
0.66
1.37
1.24
1.59
1.53
1.40
1.29
1.19
1.12
1.12
1.07
1.00
1.64
1.56
1.95
1.78
1.71
1.62
1.58
1.51
1.48
1.46
1.41
1.95
1.89
2.53
2.42
2.35
2.29
2.22
2.20
2.16
2.14
2.12
2.59
2.51
3.21
3.11
3.05
2.94
2.89
2.86
2.83
2.82
2.81
3.20
3.24
3.54
3.45
3.36
3.31
3.31
3.27
3.25
3.22
3.25
3.51
3.50
3.78
3.76
3.59
3.53
3.46
3.44
3.48
3.48
3.42
3.74
3.74
0.51
0.92
1.28
1.98
2.62
2.98
3.23
Source: Author's calculations.
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Table 2
Method
Window
5%
RADF P e r c e n t i l e s
10%
20%
50%
80%
90%
95%
ACE
9
1.89
2.07
2.39
3.05
3.58
3.87
4.11
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
J v AS
14
19
24
29
34
39
44
49
Variable
Variable
1.69
1.57
1.53
1.42
1.43
1.35
1.28
1.28
1.21
1.18
1.98
1.84
1.75
1.66
1.56
1.51
1.48
1.50
1.47
1.47
(b)
2.19
2.83
2.09
2.73
2.04
2.61
1.89
2.49
1.80
2.41
1.75
2.38
1.73
2.34
1.69
2.29
1.79
2.37
1.70
2.30
p = 0.333
3.42
3.31
3.22
3.23
3.21
3.14
3.07
2.95
2.95
7.91
3.77
3.66
3.62
3.65
3.51
3.37
3.31
3.26
3.28
3.37
4.04
3.88
3.85
3.88
3.80
3.72
3.57
3.46
3.61
3.50
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
9
14
19
24
29
34
39
44
49
Variable
1.88
1.71
1.59
1.48
1.42
1.37
1.31
1.26
1.22
1.29
1.15
2.09
1.92
1.80
1.67
1.58
1.54
1.53
1.43
1.42
1.52
1.43
2.34
2.19
2.12
2.02
1.92
1.86
1.82
1.77
1.72
1.84
1.73
7.39
3.45
3.30
3.22
3.11
3.08
3.00
3.02
2.95
2.93
3.01
7.88
3.85
3.60
3.54
3.43
3.33
3.27
3.30
3.27
3.22
3.37
3 -35
3.98
3.91
3.85
3.69
3.58
3.52
3.55
3.48
3.36
3.65
3.53
1.95
2.61
3.02
3.23
Variable
(c) 8
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
ACE
AVAS
OLS
Source:
9
14
19
24
29
34
39
44
49
Variable
Variable
=
1.48
1.38
1.28
1.25
1.22
1.12
1.12
1.09
1.05
1.38
1.36
1.65
1.51
1.49
1.43
1.38
1.40
1.35
1.33
1.30
1.61
1.56
1.89
1.80
1.74
1.68
1.65
1.61
1.60
1.58
1.56
1.89
1.83
0.36
0.79
1.21
Author's calculations.
2.89
2.71
2.63
2.53
2.45
2.38
2.28
2.24
2.24
2.40
3
www.clevelandfed.org/research/workpaper/index.cfm
Table 3
ADF and RADF as a Linear Cointegration Test
(a) ADF Percentiles
No. of
Regressors
1
1%
-0.22
5%
10%
20%
50%
80%
90%
95%
99%
0.53
0.89
1.29
1.95
2.60
2.96
3.29
3.82
(b) RADF Percentiles
Source:
Author's calculations.
The most interesting results are those for P
= 3.
In this
case there is considerable correlation between @(xt) and 8(yt),
even though they are not cointegrated.
null hypothesis in practice.
When P
This is the most likely
= 3,
the higher percentiles
(80, 90, and 95) of the ADF are about 0.1 greater than the
corresponding RADF percentiles.
The upper percentiles of the two
statistics for OLS (table 3) and the variable window procedures are
even closer. Looking at the fixed window results, it appears that
for window sizes of one-fourth to one-half the sample size, the
higher percentiles fall between those found in lines 1 and 2 of
table 3, panel a.
The critical values for the OLS ADF with two
regressors thus provide a conservative test for the ADF and RADF
when ACE with a fixed window smoother is used.
For the variable
window procedures, adding 0.2 (for an ADF test) or 0.1 (for an RADF
www.clevelandfed.org/research/workpaper/index.cfm
test) to these same critical values gives a test of about the right
size.
V.
Applications
The estimation and testing techniques of sections I11 and IV
were applied to two bivariate data sets: (i) monthly observations
of prices and dividends of the Standard
&
Poor's common stock
composite index from January 1957 through February 1990 and (ii)
quarterly U.S. nominal GNP and M2 money supply from 1959:IQ through
1989:IVQ.
For each data set, the first variable was regressed on
the second using OLS, ACE, and AVAS.
The present value model maintains that the price of a stock is
the discounted sum of expected future dividends; that is,
If dividends, d,, follow a difference-stationary process and the
discount rate, Pt, is less than one and constant, then Campbell and
Shiller (1986) argue that equation (6) implies cointegration of
dividends and prices.
using
the notation
To see why, rewrite it as
.
4 d t h = (dth - dt)
Since
4 d t follows a
stationary process, goes the argument, so too does its expectation.
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A
discounted sum of stationary variables is also stationary, so
(D
C p h ~ , ~ d tis
, stationary and p,, d, are cointegrated.
Unfortunately, the argument that the stationarity ofA,,d,
guarantees
the
stationarity
of
E,A,,dth is
incorrect.
The
expectation can change each period due to influences on agents'
expectations that are not stationary.
The argument does hold if
the optimal forecast E,A,,d,, is a linear function of past values of
p, and d, with constant coefficients.
But as seen in table 4, a
unit root cannot be rejected in the residual from a regression of
prices on dividends.
The low Durbin-Watson statistic indicates
that this is a spurious regression of the kind discussed in Granger
and Newbold (1974), and the values of the ADF and RADF statistics
are not nearly large enough to reject the hypothesis of a unit root
in the OLS residuals. Figure l(a) is a scatterplot of stock prices
and dividends with the regression line superimposed.
behavior of the residual is evident.
The LMM
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Table 4
Procedure
Series
Stock Prices and Dividends
ADF(4)
RADF (4)
DW
R~
OLS
.850
2.46
pt
5.50
-2.18
dt
Gt
ACE
-1.23
.56
-1.98
.038
.984
0(~t)
1.43
-1.23
AVAS
0 (P,)
9 (dt)
+
Source:
Author's calculations.
Figures 1(b) and 1(c) show the transformations of stock prices
and dividends estimated by the variable window ACE, while figures
l(d)
and l(e)
show the AVAS transformations.
The dividend
transformation looks similar for both procedures, but the AVAS
price transformation shows some evidence of nonlinearity not
present in the corresponding ACE transformation.
The reason for
the difference is evident in plots of the ACE and AVAS residuals,
figures l(f) and l(g).
The ACE residual variance shows a clear
trend that the AVAS price transformation has eliminated.
The DW is low for both the ACE and AVAS residuals, but the ADF
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23
and RADF statistics are well above the 95th percentiles noted in
tables 1 and 2.
This suggests that prices and dividends are
cointegrated after an appropriate transformation has been applied
to dividends.
Upon closer examination of the original series,
however, it becomes apparent that the nonlinearity in the dividend
transformation, particularly the flat spot between d
=
3 and d
=
7,
is almost entirely due to the behavior of the two series over the
1970s.
In January 1967, prices and dividends were $84.45 and
$2.96, respectively.
Fifteen years and seven months later,
dividends had risen by 155 percent (to $7.56), while stock prices
had climbed only 40 percent (to $117.86).
series have trended mostly upward.
Since that period, both
Because there appears to be
little likelihood that dividends will ever again be in the $3.00 to
$7.50 range, there is no way to tell the difference between
nonlinear cointegration and linear cointegration with time-varying
parameters for these series.
Given the well-known problems
resulting from inappropriate detrending of 1(1) time series, ACE
and AVAS transformations of trending series should be interpreted
with caution.
Economically,
the
dividend
transformation
is
not
very
satisfying. The present value theory implies cointegration between
prices and dividends, not transformations of prices and dividends.
The cause of economic understanding would be better served through
an exploration of why cointegration is not found in the data.
An
obvious starting point would be to allow for time variation in the
discount rate.
Another explanation may be that investors in the
www.clevelandfed.org/research/workpaper/index.cfm
1970s thought dividend payouts were unsustainably high, perhaps due
to the
inadequate adjustment of depreciation allowances for
inflation or obsolescence.
Some support for the latter view is
found by Campbell and Shiller, who report that the dividend-price
ratio Granger causes dividends.
A second example clearly shows the differences between ACE and
AVAS.
Engle and Granger (1987) report that GNP and M2 are
cointegrated in logarithms.
Running either ACE or AVAS on the
logarithms of the series results in transformations (figures 2[a]
through 2[d])
that appear linear.
However, if ACE and AVAS are
used with the levels of M2 and GNP (figures 2[e] through 2[h]),
only the AVAS algorithm finds the log transformation.
The ACE
algorithm finds that the very strong linear relationship it starts
out with improves only slightly on subsequent iterations, so it
stops.
There is, however, an exponential trend in the residual
variance.
AVAS tries to eliminate it, and the resulting variance-
stabilizing transformations look very much like scaled logarithms.
Table 5 shows some statistics from OLS, ACE, and AVAS.
The OLS
results are for the logs of M2 and GNP, but the others are not.
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Table 5
Procedure
Series
ADF(4)
GNP and M2
RADF ( 4)
DW
R~
-
.99834
OLS
109(yt)
109 (mt
&
-
3.90
3.98
.70
-3.37
-3.44
.OO
.15
ACE
0 (yt)
@ (mt
t
AVAS
0 (Yt)
@ (mt)
6-
Source:
VI.
Author's calculations.
Conclusion
Attractor sets are the special case of nonlinear cointegration
in which the cointegrating function is the Euclidean distance
function.
However, series can be nonlinearly cointegrated in an
economically interesting way without having an attractor.
It may
be better to aim future research at methods of discovering
interesting cointegrating functions rather than at looking for
attractors.
If several series are cointegrated only after they are
individually nonlinearly transformed, this can be thought of as an
additively separable cointegrating function.
Granger and Hallman
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26
(1990) propose using ACE to estimate the transformations and the
ADF to test for nonlinear cointegration.
In this paper, it appears
that a version of ACE modified to stabilize the residual variance
may
be
more
useful.
Once
the
possibility
of
nonlinear
transformations of the data is acknowledged, it would be sensible
to employ a unit root test that is robust to such changes.
The
RADF is expressly designed for this purpose, so both it and the
conventional ADF are employed.
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References
L. Breiman and J. H. Friedman, "Estimating Optimal
Transformations for Multiple Regression and CorrelationtVv
Journal of the American Statistical Association, vol. 80, pp.
580-97, 1985.
J. Y. Campbell and R. J. Shiller, l1Cointegrationand Tests of
Present Value Models, Working Paper No. 1885, National Bureau
of Economic Research, 1986.
W. S. Cleveland, rlRobustLocally Weighted Regression and
Smoothing Scatterplots,~Journal of the American Statistical
Association, vol. 74, pp. 829-36, 1979.
R. F. Engle and C. W. J. Granger, "Cointegration and Error
Correction:
Representation, Estimation
and
Testing,"
Econometrica, vol. 55, pp. 251-76, 1987.
R. F. Engle and B. S. Yoo, "Forecasting and Testing in
Cointegrated Systems," Journal of Econometrics, vol. 35, pp.
143-59, 1987.
J. H. Friedman and W. Stuetzle, NSmoothing of Scatterplots,It
Technical Report ORIONOO6, Stanford University, Department of
Statistics, 1982.
C. W. J. Granger and J. J. Hallman, "The Algebra of I(1) ,Iv
Finance and Economics Discussion Series, Board of Governors of
the Federal Reserve System, 1988.
C. W. J. Granger and J. J. Hallman, "Long Memory Series with
AttractorstVvOxford Bulletin of Economics and Statistics,
forthcoming, 1990.
C. W. J. Granger and P. Newbold, llSpuriousRegressions in
Econometrics, Journal of Econometrics, vol 2, pp. 111-20,
1974.
.
J. C. Marhoul and A. B. Owen, llConsistencyof Smoothing with
Running Linear Fits,I1 Technical Report LCS 008, Stanford
University, Department of Statistics, 1984.
P. C. B. Phillips, "Time Series Regression with a Unit Root,"
Econometrica, vol. 55, pp. 277-301, 1987.
B. W. Silverman, I1Some Aspects of the Spline Smoothing
Approach to Non-parametric Regression Curve Fitting (with
discu~sion),~~
Journal of the Royal Statistical Society, vol.
47, pp. 1-52, 1985.
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13.
R. Tibshirani, I1EstimatingTransformations for Regression via
~
of the
Additivity and Variance S t a b i l i ~ a t i o n , ~Journal
American Statistical Association, vol. 83, pp. 394-405, 1988.
14.
B. S. Yoo, "Co-Integrated Time Series: Structure, Forecasting
and Testing," unpublished Ph.D. dissertation, University of
California, San Diego, 1987.
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Figure I (b): ACE Transformation of Stock Prices
stock.prices
Figure I (c): ACE Transformation of Stock Dividends
Source:
Author's c a l c u l a t i o n s .
www.clevelandfed.org/research/workpaper/index.cfm
Figure 1(d): AVAS Transformation of Stock Prices
stock.prices
Figure 1(e): AVAS Transformation of Stock Dividends
Source:
Author's calculations .
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Figure 1(f): ACE Residual for Prices and Dividends
1960
1970
1980
Figure 1(g): AVAS Residual for Prices and Dividends
1960
Source: Author's calculations.
1970
1990
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Figure 2(a): ACE Transformation of log(rn2.q)
log(m2.q)
Figure 2(b): ACE Transformation of log(gnp.q)
Source:
Author's calculations.
www.clevelandfed.org/research/workpaper/index.cfm
Figure 2(c): AVAS Transformation of log(m2.q)
Figure 2(d): AVAS Transformation of log(gnp.q)
LC!
7-
LC!
7I
Source: Author's calculations.
www.clevelandfed.org/research/workpaper/index.cfm
Figure 2(e): ACE Transformation of m2.q
-
.
..
'0
8
.
.
.
.**
*.
**
-
. .
.**
0.
Z
-
.,
*"
***
.
.*
.*
**
.*.
.**
.***
04.
-
/*@@
500
1000
I
I
I
I
1500
2000
2500
3000
m2.q
Figure 2 0 : ACE Transformation of gnp.q
Source:
Author's c a l c u l a t i o n s .
www.clevelandfed.org/research/workpaper/index.cfm
Figure 2(g): AVAS Transformation of m2.q
Solid line is scaled log transform
500
1000
1500
2000
2500
m2.q
Figure 2(h): AVAS Transformation of gnp.q
Solid line is scaled log transform
Author's calculations .
3000
@FedFRASER