From Unit Root To Cointegration

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From Unit Root To Cointegration

• Putting Economics into Econometrics

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Many (perhaps most) macroeconomic variables are
non stationary Many of these are difference
stationary, I(1), variables.

• Economics - many variables have stable long-run
  relationships.
• Consumption - Income
• Prices - Wages
• Prices at home - prices abroad

We can use the unit root testing techniques to
identify variables which have stable long-run
relationships with one another (as opposed to
spurious regression)
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Example Are Y (disposable income) and X
(consumption) cointegrated? First be satisfied
that the two time series are I(1). E.g. apply
unit root tests to X and Y in turn.
They are clearly not stationary But they seem to
move together
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Economic theory tells us that there should be (at
least in the long run) a relation like
A regression between these variables gives
EQ( 1) Modelling C by OLS (using Lecture6a.in7)
The estimation sample is 1 to 99
Coefficient Std.Error t-value t-prob
Part.R2 Constant 4.85755 0.1375
35.3 0.000 0.9279 Y
1.00792 0.005081 198. 0.000
0.9975 sigma 0.564679 RSS
30.9296673 R2 0.997541
F(1,97) 3.935e004 0.000 log-likelihood
-82.8864 DW 2.28 no.
of observations 99 no. of parameters
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But is this regression spurious? Or is there a
genuine long run relationship?
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COINTEGRATION
Any linear combination of I(1) variables is
typically spurious. However if there is a
long-run relationship, errors have tendency to
disappear and return to zero i.e. are I(0). If
there exists a relationship between two non
stationary I(1) series, Y and X , such that the
residuals of the regression
are stationary, then the variables in question
are said to be cointegrated There is a Long Run
Relationship towards which they always come back
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Single Equations Errors
ut 0 - Disequilibrium errors
(i.e. ut Yt - ß0 - ß1Xt)
No tendency to return to zero
Error rarely drifts from zero
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Stationary Errors

If we have two independent non-stationary
series, then we may find evidence of a
relationship when none exists (i.e. spurious
regression problem). One way to test if there
is a relationship between non- stationary data
is if disequilibrium errors return to zero. If
long run relationship exists then errors should
be a stationary series and have a zero mean.
ut 0
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Our regression between consumption and income
How can we distinguish between a genuine long-run
relationship and a spurious regression? We need a
test.
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Testing for Cointegration
After estimating the model save residuals from
static regression. (In PcGive after running
regression click on Test and Store Residuals)
Informally consider whether stationary.
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(1) Cointegrating Regression Durbin Watson
(CRDW) Test At 0.05 per cent significance level
with sample size of 100, the critical value is
equal to 0.38. Ho DW 0 gt no cointegration
(i.e. DW stat. is less than 0.38) Ha DW gt 0 gt
cointegration (i.e. DW stat. is greater than
0.38) Ho ut ut-1 et Ha ut ?ut-1
et ? lt 1 N.B. Assumes that the
disequilibrium errors ut can be modelled by a
first order AR process. Is this a valid
assumption? May require a more complicated
model.
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First Test Cointegrating Regression Durbin
Watson Test (CRDW)
If the residual are non stationary, DW will go to
0 as the sample size increases. So large values
of DW are taken as evidence for rejection of the
null hypothesis of NO COINTEGRATION
EQ( 1) Modelling Y by OLS (using Lecture
6a.in7) The estimation sample is 1 to
99 Coefficient Std.Error
t-value t-prob Part.R2 Constant
4.85755 0.1375 35.3 0.000 0.9279 X
1.00792 0.005081 198.
0.000 0.9975 sigma 0.564679
RSS 30.9296673 R2
0.997541 F(1,97) 3.935e004
0.000 log-likelihood -82.8864 DW
2.28 no. of observations
99 no. of parameters 2 CRDW test
statistic 2.28 gtgt 0.38 5 critical
value. This suggests cointegration - assumes
residuals follow AR(1) model.
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Second Test Cointegrating Regression DF test
(CRDF)
1 - Perform the cointegrating regression
2 Save the residuals, 3 Run auxiliary
regression
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Cointegrating Regression Dickey Fuller (CRDF)
Test ?ut f ut-1 et Critical
Values (CV) are from MacKinnon (1991) Ho f
0 gt no cointegration (i.e. TS is greater
than CV) Ha f lt 0 gt cointegration (i.e.
TS is less than CV)
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Cointegrating Regression Dickey Fuller (CRDF)
Test BETTER Use lagged differenced terms to
avoid serial correlation. ?ut f ut-1
?1?ut-1 ?2?ut-2 ?3?ut-3 ?4?ut-4 et Use
F-test of model reduction and also minimize
Schwarz Information Criteria. Critical
Values (CV) are from MacKinnon (1991) Ho f
0 gt no cointegration (i.e. TS is greater
than CV) Ha f lt 0 gt cointegration (i.e.
TS is less than CV)
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Testing for Cointegration

Using CRDF we incorporate lagged dependent
variables into our regression ?ut f
ut-1 ?1?ut-1 ?2?ut-2 ?3?ut-3 ?4?ut-4
et And then assess which lags should be
incorporated using model reduction tests and
Information Criteria. Progress to date Model
T p log-likelihood SC
HQ AIC EQ( 2) 94 5 OLS
-74.238306 1.8212 1.7406
1.6859 EQ( 3) 94 4 OLS
-74.793519 1.7847 1.7202 1.6765 EQ(
4) 94 3 OLS -74.797849
1.7364 1.6881 1.6553 EQ( 5) 94
2 OLS -74.948145 1.6913 1.6591
1.6372 EQ( 6) 94 1 OLS
-75.845305 1.6621 1.6459
1.6350 Tests of model reduction (please ensure
models are nested for test validity) EQ( 2) --gt
EQ( 6) F(4,89) 0.77392 0.5450 EQ( 3) --gt
EQ( 6) F(3,90) 0.67892 0.5672 EQ( 4) --gt
EQ( 6) F(2,91) 1.0254 0.3628 EQ( 5) --gt
EQ( 6) F(1,92) 1.7730 0.1863
Consequently we choose ?ut f ut-1 et
All model reduction tests are accepted hence move
to most simple model
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Testing for Cointegration using CRADF test
Step 1 Estimate cointegrating regression
Yt ß0 ß1Xt ut EQ( 1) Modelling Y by
OLS (using Lecture 6a.in7) The estimation
sample is 1 to 99 Coefficient
Std.Error t-value t-prob Part.R2 Constant
4.85755 0.1375 35.3 0.000
0.9279 X 1.00792 0.005081
198. 0.000 0.9975 sigma
0.564679 RSS 30.9296673 R2
0.997541 F(1,97) 3.935e004
0.000 log-likelihood -82.8864 DW
2.28 no. of observations
99 no. of parameters 2 And save
residuals.
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Testing for Cointegration
Step 2 Use estimated in the auxiliary
regression model ?ut fut-1 et EQ( 6)
Modelling dresiduals by OLS (using
Lecture6a.in7) The estimation sample is 6
to 99 Coefficient Std.Error
t-value t-prob Part.R2 residuals_1
-1.16140 0.1024 -11.3 0.000
0.5805 sigma 0.545133 RSS
27.6367834 log-likelihood -75.8453
DW 1.95 Which means ?ut
-1.161 ut-1 et (-11.3) CRDF test
statistic -11.3 ltlt -3.39 5 Critical Value
from MacKinnon. Hence we reject null of no
cointegration between X and Y.
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Testing for Cointegration
Advantages of (CRDF) Test Engle and Granger
(1987) compared alternative methods for testing
for cointegration. (1) Critical values depend
on the model used to simulated the data. CRDF
was least model sensitive. (2) Also CRDF has
greater power (i.e. most likely to reject a false
null) compared to the CRDW test.
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Testing for Cointegration
Disadvantage of (CRDF) Test - Although the
test performs well relative to CRDW test there
is still evidence that CRDF have absolutely low
power. Hence we should show caution in
interpreting the results.
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Cointegration and Consistency

OLS estimates with I(0) variables are said to be
consistent. As the sample size increases they
converge on their true value. However if
the true relationship between variables includes
dynamic terms Yt ?0 ?1Xt ?2Yt-1
?3Xt-1 ut Static models estimated by OLS will
be bias or inconsistent. Yt ß0 ß1Xt
ut Stock (1987) found that if Yt and Xt are
cointegrated then OLS estimates of ß0 and ß1 will
be consistent.
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Cointegration and Superconsistency
Indeed, Stock went further and suggested that
estimated coefficients from cointegrated
regressions will converge at a faster rate than
normal. i.e. super consistent. Coefficients
from a cointegrated regression are super
consistent. gt (i) simple static
regression dont necessarily give spurious
results. (ii) dynamic misspecification is not
necessarily a problem. Consequently we can
estimate simple regression Yt ß0 ß1Xt
ut even if there are important dynamic
terms Yt ?0 ?1Xt ?2Yt-1 ?3Xt-1 ut
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Cointegration and Superconsistency
However, superconsistency is a large sample
result. Coefficients may be biased in finite
samples (i.e. typical sample periods) due to
omitted lagged values of Yt and Xt Bias in
static regressions is related to R2 . A high
R2 indicates that the bias will be smaller.
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Testing for Cointegration Summary
To test whether two I(1) series are cointegrated
we examine whether the residuals are I(0). (a)
We firstly use informal methods to see if they
are stationary (1) plot time series of
residuals (2) plot correlogram of
residuals (b) Two formal means of testing for
cointegration. (1) CRDW - Cointegrating
Regression Durbin Watson Test (2) CRDF -
Cointegrating Regression Dickey Fuller Test