Lecture 27:
Cointegration
(Chapter 18.6–18.7)
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Agenda
• Review
• Common Trends (Chapter 18.6)
• Cointegration (Chapter 18.6)
• The Drunk and Her Dog (Chapter 18.6)
• Dynamic OLS (Chapter 18.7)
• Example: Deficits (Chapter 18.7)
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27-2
TABLE 18.1 The Consistency of OLS and the
Validity of Conventional Tests in the Face of
Stochastic Trends*
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27-3
Review
• When both e and Z contain stochastic
trends, our regression is spurious.
• OLS often misleads us into believing
Y and Z are related, even when they
are not.
• We need a test for stochastic trends,
and an estimation procedure for
variables exhibiting stochastic trends.
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27-4
Review of Unit Root Tests
• The desired test:
Zt   Zt -1   0  1t  vt
H 0 :   1 vs. H a :   1
• The problem:
– Under the null hypothesis, OLS is not
asymptotically normal.
• The solution:
– Develop a new test statistic whose distribution
we know.
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27-5
Review of Unit Root Tests (cont.)
• Dickey and Fuller developed a suitable
test. They needed to modify the
regression slightly:
Zt  Zt 1  (   1)Zt 1   0  1t  vt
• To test  = 1, we test that the
coefficient on Zt-1 is 0 against the
1-sided alternative that it is less than 0.
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27-6
Review of Unit Root Tests (cont.)
• We cannot use our usual t-statistic
critical values because the Dickey–
Fuller asymptotic distribution
is skewed.
• Dickey and Fuller determined the
appropriate critical values.
• The critical values depend on whether
the deterministic drift term is included.
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27-7
Review of Unit Root Tests (cont.)
Zt  Zt 1  (   1)Zt 1   0  1t  vt
• The Dickey–Fuller test assumes that
the vt are serially uncorrelated.
• If we are concerned that the vt are
serially correlated, we need to use the
Augmented Dickey–Fuller (ADF) test.
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27-8
Review of Unit Root Tests (cont.)
• The Augmented Dickey–Fuller test adds
lagged DZ values to the explanators.
DZt  Zt  Zt 1
 (   1) Zt 1   0  1t   2 DZt 1   3DZt  2  vt
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27-9
Review of Unit Root Tests (cont.)
• Caution: failing to reject the null of a
stochastic trend does NOT establish the
presence of a stochastic trend.
• The Dickey–Fuller test often has
weak power.
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27-10
Review
• The Granger and Newbold strategy:
• Both Y and Z contain stochastic trends
(for example, let them both follow
random walks).
• Focus on CHANGES in Y and Z,
differencing out the stochastic trend.
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27-11
Review (cont.)
t
Yt  Yt 1  vt   vs (Y does not depend on Z )
s 0
DYt  Yt  Yt 1  vt
t
Z t  Z t 1  t   vs
s 0
DZ t  Z t  Z t 1  t
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27-12
Review (cont.)
t
Yt  Yt 1  vt   vs
s 0
t
Z t  Z t 1  t   vs
s 0
If we regress DYt  1DZ t  t
then t  vt
Neither the explanator nor the error term contain
stochastic trends. The regression satisfies the
Gauss–Markov conditions.
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27-13
Review (cont.)
• If Y and Z BOTH receive a large positive
shock, then DY and DZ will both be
above their means.
• However, in the next period, the shock
will no longer be evident in DY and DZ.
• DY and DZ revert to the mean.
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27-14
Review (cont.)
• Danger: if Y or Z does NOT contain a
stochastic trend, we do NOT want
to difference.
• DY has less variation than Y, hurting
efficiency.
• Differencing can induce serial
correlation.
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27-15
Review (cont.)
• The Dickey–Fuller test warns of the
possibility of a stochastic trend.
• The Newbold–Granger method is
unattractive in the absence of a
stochastic trend.
• Tests from differenced regressions tend
to have lower power.
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27-16
Review (cont.)
• Variables with no stochastic trends are
Integrated of Order Zero, written I(0).
• The variables we have studied
previously in this course are I(0).
• The “white noise” innovations in a
random walk are I(0).
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27-17
Review (cont.)
• Variables with stochastic trends, but
whose first differences contain no
stochastic trends, are Integrated of
Order One, written I(1).
• Variables that follow a random walk
are I(1).
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27-18
Review (cont.)
• When both e and Z contain stochastic
trends, our regression is spurious.
• OLS often misleads us into believing
Y and Z are related, even when they
are not.
• When using a stochastically trending
explanator, we can use a Dickey–Fuller
test on the residuals.
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27-19
Common Trends (Chapter 18.6)
• Suppose Yt  0  1 X t  ut
• The Dickey–Fuller test can fail to reject
the null hypothesis that Y contains a
stochastic trend.
• A stochastic trend in Y can come from a
stochastic trend in X, a stochastic trend
in u, or both.
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27-20
Common Trends (cont.)
• Suppose Yt  0  1 X t  ut
• We have seen that when both X and u
contain stochastic trends, our
regression is spurious: we are far too
likely to reject the null that 1 = 0, even
if it IS zero.
• But what if 1 really is non-zero?
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27-21
Common Trends (cont.)
• Suppose Yt  0  1 X t  ut
• If 1 really is non-zero, and X contains a
stochastic trend, then Y inherits that
stochastic trend from X.
• Moreover, Y and X share that same
trend in common.
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27-22
Figure 18.9 Real per Capita GDP and
Real per Capita Consumption, 1948–1998
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27-23
Common Trends
• Suppose Yt  0  1 Xt  ut
• If 1 really is non-zero, and X contains
a stochastic trend, then Y inherits that
stochastic trend from X.
• If u ALSO contains a stochastic
trend, then Y might contain several
stochastic trends.
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27-24
Cointegration (Chapter 18.6)
• Variables that contain stochastic trends
exhibit increasing variance over time.
• Such variables can get very far away
from their ex ante expectations.
• However, they may stay relatively
close together.
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27-25
Cointegration (cont.)
Yt   0  1 X t  ut
• If Y and X both contain stochastic
trends, and u does not, then Y inherits
its only stochastic trending component
from X.
• We call Y and X cointegrated.
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27-26
Cointegration (cont.)
• Cointegrated variables both wander
stochastically, but stay near each other.
• They adjust to each other’s locations
through a process of error correction.
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27-27
Cointegration (cont.)
• Note: cointegration also extends to the
multiple regression case.
• The key is that the disturbance term
does not contain a stochastic trend.
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27-28
Cointegration (cont.)
• When there is a linear combination of
trending Y ’s and X ’s that itself contains
no variable trend, we call the variables
“cointegrated.”
• i.e., if Yt   0  1 X1t   2 X 2t  ut
and Y, X1, and X2 contain variable
trends, but u does not, we say Y, X1,
and X2 are cointegrated.
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27-29
Cointegration (cont.)
• Consider:
Yt  0  1 X t  ut
where both Y and X contain variable trends,
but u does not, so Y and X are cointegrated.
• Question: can we estimate 1 by analyzing
changes in Y and X (the Newbold–Granger
method)?
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27-30
The Drunk and Her Dog
• As an example, consider the case of a
drunk and her dog.
• Both the drunk and the dog, on their
own, might tend to wander aimlessly.
• However, the dog might make sure to
stay close to its drunken master.
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27-31
The Drunk and Her Dog (cont.)
• The instructive tale of the drunk and
her dog…
– To estimate 1 in: Yt  0  1Zt  ut
should we study
(Yt - Yt -1 )  DYt  1DZt  ut - ut -1?
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27-32
The Drunk and Her Dog (cont.)
(Yt - Yt -1 )  DYt  1DZt  ut - ut -1
• If two stochastically-trending variables
are truly related, a specification solely in
changes is mis-specified—it is missing
the cointegrating relationship(s) among
the variables.
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27-33
The Drunk and Her Dog (cont.)
• The Lesson:
– If we are going to look at changes in Y
and X when Y and X are cointegrated,
we must also examine the “cointegrating
vector” that will be the error correction
mechanism holding together Y and X as
they trend.
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27-34
The Drunk and Her Dog (cont.)
• In the example, the drunk and her dog
both wander aimlessly.
• However, they periodically engage in
error correction to move closer together.
• Neither one is following a random walk.
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27-35
The Drunk and Her Dog (cont.)
• Consider four possible simulated paths
for the drunk and her dog.
• The vertical axis measures how far
they’ve each wandered from the
local pub.
• The horizontal axes tracks their steps,
from 1 to 1,0000.
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27-36
Figure 18.11 Four Paths for the Drunk
and Her Dog (1 of 2)
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27-37
Figure 18.11 Four Paths for the Drunk
and Her Dog (2 of 2)
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27-38
Figure 18.11 Four Paths for the Drunk
and Her Dog
27-39
The Drunk and Her Dog (cont.)
• The order of cointegration depends
on the trend in the distance between
Y and Z.
• If the distance between Y and Z contains
no stochastic trend, then Y and Z are
cointegrated of order 0.
• For Y and Z to be cointegrated of order r,
then both Y and Z are integrated I(r+1).
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27-40
The Drunk and Her Dog (cont.)
• We can model the drunk’s and the
dog’s cointegrated motion as:
Yt  Yt 1  e t  a1 (Yt 1  Z t 1 )
Z t  Z t 1  wt  a 2 (Yt 1  Z t 1 )
• a1 and a2 are the speeds of
adjustment.
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27-41
The Drunk and Her Dog (cont.)
Yt  Yt 1  e t  a1 (Yt 1  Z t 1 )
Z t  Z t 1  wt  a 2 (Yt 1  Z t 1 )
• The second terms on the RHS of
each equation are error correction
mechanisms.
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27-42
The Drunk and Her Dog (cont.)
• In general, cointegration does NOT
require that the Yt – Zt contain no
stochastic trend.
• We need only some linear combination
of the variables to contain no
stochastic trend.
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27-43
A More General Error Correction Model
Long-Run Relationship
Yt  1 X t  t
Short-Run Error Correction
DYt  a 0   y (Yt 1  1 X t 1 )  t
DX t  0   x (Yt 1  1 X t 1 )  vt
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27-44
A More General
Error Correction Model (cont.)
DYt  a 0   y (Yt 1  1 X t 1 )  t
DX t  0   x (Yt 1  1 X t 1 )  vt
Yt  1 X t  t
• Y and X adjust towards their long-run
relationship at speeds y and x.
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27-45
A More General
Error Correction Model (cont.)
DYt  a 0   y (Yt 1  1 X t 1 )  t
DX t  0   x (Yt 1  1 X t 1 )  vt
Yt  1 X t  t
• The  (Yt-1 – 1Xt-1) terms are error
correction mechanisms to keep Y and
X close to their long-run relationship.
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27-46
A More General
Error Correction Model (cont.)
In the presence of serial correlation, we need to
include lagged differences in the variables.
DYt  a 0  a1DYt 1  a 2 DX t 1   y (Yt 1  1 X t 1 )  t
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27-47
Dynamic OLS (Chapter 18.7)
• When et contains a stochastic trend,
OLS is inconsistent. Use the Granger
and Newbold method (differencing Y
and X ).
• When the et ’s do not contain stochastic
trends, but an explanator does, OLS is
super consistent, but not asymptotically
normal. Our tests do not work (for the
trending explanator).
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27-48
Dynamic OLS (cont.)
Yt  0  1Z1t  2 Z2t  3 X t  e t
• Suppose Y, Z1, and Z2 contain stochastic
trends, but X and e do not.
• We can estimate all the coefficients
consistently.
• Only our estimate of 3 is asymptotically
normal.
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27-49
Dynamic OLS (cont.)
Yt  0  1Z1t  2 Z2t  3 X t  e t
• Our hypothesis tests are not asymptotically
valid for the variables that contain a
stochastic trend (Z1 and Z2).
• Stock and Watson’s insight: change the
equation so that the 1 and 2 coefficients
apply to non-trending variables.
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27-50
Dynamic OLS (cont.)
• Dynamic OLS:
– A method for estimating cointegrated
models
– Respecify the model so that the coefficients
of interest apply to non-trending variables.
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27-51
Dynamic OLS (cont.)
Case: no serial correlation
We wish to estimate
Yt   0  1Z1t   2 Z 2t   3 X t  e t
Yt , Z1t , Z 2t contain stochastic trends.
Rewrite the model as
Yt   0  1Z1t   2 Z 2t   3 X t   4 DZ1t   5 DZ 2t  t
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27-52
Dynamic OLS (cont.)
Case: no serial correlation
Rewrite the model as
Yt   0  1Z1t   2 Z 2t   3 X t   4 DZ1t   5 DZ 2t  t
After controlling for DZ1t , there is no stochastic
trend in Z1t .
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27-53
Dynamic OLS (cont.)
• Serial correlation complicates
Dynamic OLS.
• We must add not only DZ but also leads
and lags of DZ.
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27-54
Dynamic OLS (cont.)
Case: serial correlation
We wish to estimate
Yt   0  1Z1t   3 X t  e t
Yt and Z1t contain stochastic trends.
Rewrite the model as
Yt   0  1Z1t   2 X t   1DZ1,t  2
  2 DZ1,t 1   3 DZ1,t   4 DZ1,t 1
  5 DZ1,t  2  t
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27-55
Dynamic OLS (cont.)
• Dynamic OLS allows us to draw
inferences about relationships among
stochastically trending variables.
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27-56
Example: Deficits (Chapter 18.7)
• Example: Interest Rates and Deficits
• Do large Federal budget deficits drive
up long-term interest rates?
• We want to regress the 10 year treasury
bond rate on the 1 year rate, inflation,
the real deficit per capita, and the
change in real per capita income.
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27-57
Example: Deficits (cont.)
• Augmented Dickey–Fuller tests suggest
that the interest rates, inflation, deficit
per capita, and level of income per
capita are all integrated I(1).
• Our regression may be consistent,
depending on whether the disturbances
contain a stochastic trend.
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27-58
TABLE 18.12 OLS Estimates of a Model
of Long-term Interest Rates
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27-59
Example: Deficits
• The Durbin–Watson statistic of 1.71
suggests that disturbances are not
serially correlated.
• We apply a Dickey–Fuller test to the
residuals to test for a stochastic trend in
the disturbances.
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27-60
TABLE 18.13 Augmented Dickey–Fuller Test
for a Stochastic Trend in Disturbances
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27-61
Example: Deficits
• The Dickey–Fuller test rejects the
null hypothesis of a stochastic trend in
the disturbances.
• We do NOT seem to be in the spurious
regression case.
• Instead, we are in the cointegration case.
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27-62
Example: Deficits (cont.)
• Because Deficits and 10-year interest
rates are cointegrated, our coefficient on
USDEF is consistent.
• Our t-test on USDEF is incorrect.
• We can re-estimate the analysis
with Dynamic OLS to obtain proper
test statistics.
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27-63
TABLE 18.14
Dynamic
OLS Estimates
of a Model of
Long-term
Interest Rates
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27-64
Example: Deficits
• The Dynamic OLS t-statistic on
USDEF is 5.59
• We can reject the null hypothesis that
USDEF = 0
• Deficits appear to raise interest rates
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27-65
Example: Deficits (cont.)
• An error correction specification,
including the speed of adjustment ,
provides richer information about the
short-run behavior of interest rates.
• We can use the (consistent) coefficients
from the Dynamic OLS model to
construct estimated distances, so we
can estimate the speed of adjustment.
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27-66
Example: Deficits (cont.)
1) Use Dynamic OLS to estimate 1 ,  2 ,  3
2) Use these estimates to construct
Distancet  Rate10t  ˆ1 Rate1t  ˆ2 Inf t  ˆ3USDEFt
3) Estimate the speed of error correction  using
DRatet  a 0   ( Distancet -1 )  t
Note: See the textbook for the case with
serial correlation.
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27-67
Time Trends
• What have we learned about
time trends?
• We know that many macroeconomic
variables contain stochastic trends.
• We must be on guard against spurious
regressions and cointegration.
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27-68
Time Trends (cont.)
1. Use Dickey–Fuller or ADF tests
on time series data to judge which
variables are likely to contain
stochastic trends.
2. If you use stochastically trending
variables in a regression, follow up
with a DF or ADF test on the residuals.
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27-69
Time Trends (cont.)
3. If an explanator and the disturbances both
contain stochastic trends, use the Granger
and Newbold method for spurious
regressions (difference the variables).
4. If the disturbances are not stochastically
trending, but Y is, then use dynamic OLS
and an error correction regression to
examine long-run and short-run behavior.
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27-70
Time Trends (cont.)
5. Warning: Dickey–Fuller and Augmented
Dickey-Fuller tests often have low power.
We seldom know with great confidence
whether variables are integrated.
6. In addition, regressions with differenced
variables as in the Newbold–Granger
method are also more likely to produce
low-power hypothesis tests.
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27-71
TABLE 18.1 The Consistency of OLS and the
Validity of Conventional Tests in the Face of
Stochastic Trends*
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27-72
Review
• Suppose Yt   0  1 X t  ut
• The Dickey–Fuller test can fail to reject
the null hypothesis that Y contains a
stochastic trend.
• A stochastic trend in Y can come from a
stochastic trend in X, a stochastic trend
in u, or both.
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27-73
Review (cont.)
Yt   0  1 X t  ut
• If Y and X both contain stochastic
trends, and u does not, then Y inherits
its only stochastic trending component
from X.
• We call Y and X cointegrated.
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27-74
Review (cont.)
• Cointegrated variables both wander
stochastically, but stay near each other.
• They adjust to each other’s locations
through a process of error correction.
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27-75
Review (cont.)
• The Lesson:
– If we are going to look at changes in Y
and X when Y and X are cointegrated, we
must also examine the “cointegrating
vector” that will be the error correction
mechanism holding together Y and X as
they trend.
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27-76
Review (cont.)
• The order of cointegration depends on
the trend in the distance between Y and Z.
• If the distance between Y and Z contains
no stochastic trend, then Y and Z are
cointegrated of order 0.
• For Y and Z to be cointegrated of order r,
then both Y and Z are integrated I(r + 1).
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27-77
Review (cont.)
• We can model cointegrated motion as
Yt  Yt 1  e t  a1 (Yt 1  Z t 1 )
Z t  Z t 1  wt  a 2 (Yt 1  Z t 1 )
• a1 and a2 are the speeds of
adjustment.
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27-78
Review (cont.)
Long-Run Relationship
Yt  1 X t  t
Short-Run Error Correction
DYt  a 0   y (Yt 1  1 X t 1 )  t
DX t  0   x (Yt 1  1 X t 1 )  vt
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Review (cont.)
DYt  a 0   y (Yt 1  1 X t 1 )  t
DX t  0   x (Yt 1  1 X t 1 )  vt
Yt  1 X t  t
• Y and X adjust towards their long-run
relationship at speeds y and x.
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Review (cont.)
• When et contains a stochastic trend,
OLS is inconsistent. Use the Granger
and Newbold method (differencing
Y and X ).
• When the et ’s do not contain stochastic
trends, but an explanator does, OLS is
super consistent, but not asymptotically
normal. Our tests do not work (for the
trending explanator).
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Review (cont.)
Yt   0  1Z1t   2 Z 2t  3 X t  e t
• Suppose Y, Z1, and Z2 contain stochastic
trends, but X and e do not.
• We can estimate all the coefficients
consistently.
• Only our estimate of 3 is asymptotically
normal.
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Review (cont.)
Yt   0  1Z1t   2 Z 2t  3 X t  e t
• Our hypothesis tests are not asymptotically
valid for the variables that contain a
stochastic trend (Z1 and Z2).
• Stock and Watson’s insight: change the
equation so that the 1 and 2 coefficients
apply to non-trending variables.
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Review (cont.)
• Dynamic OLS:
– A method for estimating cointegrated
models
– Respecify the model so that the coefficients
of interest apply to non-trending variables.
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Review (cont.)
Case: no serial correlation
We wish to estimate
Yt   0  1Z1t   2 Z 2t   3 X t  e t
Yt , Z1t , Z 2t contain stochastic trends.
Rewrite the model as
Yt   0  1Z1t   2 Z 2t   3 X t   4 DZ1t   5 DZ 2t  t
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Review (cont.)
Case: no serial correlation
Rewrite the model as
Yt   0  1Z1t   2 Z 2t   3 X t   4 DZ1t   5 DZ 2t  t
After controlling for DZ1t , there is no stochastic
trend in Z1t .
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Review (cont.)
• An error correction specification,
including the speed of adjustment ,
provides richer information about the
short-run behavior of interest rates.
• We can use the (consistent) coefficients
from the Dynamic OLS model to
construct estimated distances, so we
can estimate the speed of adjustment.
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Review (cont.)
Case: no serial correlation
1) Use Dynamic OLS to estimate 1 ,  2 ,  3
2) Use these estimates to construct
Distance  Y  ˆ Z  ˆ Z  ˆ Z
t
t
1 1t
2
2t
3
3t
3) Estimate the speed of error correction  using
DYt  a 0   ( Distancet -1 )  t
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Review (cont.)
• What have we learned about
time trends?
• We know that many macroeconomic
variables contain stochastic trends.
• We must be on guard against spurious
regressions and cointegration.
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Review (cont.)
1. Use Dickey–Fuller or ADF tests
on time series data to judge which
variables are likely to contain
stochastic trends.
2. If you use stochastically trending
variables in a regression, follow up
with a DF or ADF test on the residuals.
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Review (cont.)
3. If an explanator and the disturbances both
contain stochastic trends, use the Granger
and Newbold method for spurious
regressions (difference the variables).
4. If the disturbances are not stochastically
trending, but Y is, then use dynamic OLS
and an error correction regression to
examine long-run and short-run behavior.
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Review (cont.)
5. Warning: Dickey–Fuller and Augmented
Dickey–Fuller tests often have low power.
We seldom know with great confidence
whether variables are integrated.
6. In addition, regressions with differenced
variables as in the Newbold–Granger
method are also more likely to produce
low-power hypothesis tests.
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