Cointegration in Single Equations: Lecture 5

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Cointegration in Single Equations Lecture 5
Introduction Using an Error Correction Model
(ECM) assumes there is a long-run relationship
between the variables in a regression. We have
shown it isnt enough to show high
correlation. High R2 and large t-ratio for
independent variables. High correlation may be
spurious, when using non-stationary
variables. We can avoid this problem if
long-run relationship is cointegrated Concept
of cointegration introduced by Granger in
1981. Second section of lectures concerns
relationships of this type.
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Long-run Relationships
Consider the following static regression between
two variables Yt ß0 ß1Xt ut This
relationship has the disequilibrium error ut (a
linear combination of Yt and Xt) where ut
Yt - ß0 - ß1Xt Engle and Granger (1987) if a
long-run relationship exists, then the
disequilibrium error should have a tendency to
disappear. Disequilibruim error - rarely drift
far from zero - often cross the
zero line - Equilibrium will
occasionally occur
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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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Cointegration and Order of Integration
If a time series has to be differenced to
become stationary it is I(1). Any linear
combination of I(1) variables is typically
spurious. However if there is a
long-run relationship, errors have a
tendency to disappear and return to zero i.e.
are I(0). If a linear combination of two I(1)
variables generates I(0) errors, we say that
the variables are cointegrated.
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Cointegration in Single Equations
Definition Two time series are said to be
cointegrated of order d, b, written CI(d, b) if
(a) they are both integrated of order d,
I(d) and (b) there exists some linear
combination of the two series that is
integrated of order d - b, where b gt
0. Compares with spurious regressions, if two
time series are I(d), then in general any linear
combination of the two series will be I(d). That
is the residuals from regressing Yt on Xt are
I(d).
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Cointegration in Single Equations

Cointegration approach is based on two time
series which are I(1). If one is I(1) and other
is I(0) then the relationship can not be
cointegrated. Example Yt 2 Yt-1 ut
and Xt 1 0.5Xt-1 ut Yt I(1)
Xt I(0) Yt and Xt are integrated of
different orders. Yt is increasing in time while
Xt is constant. Distance between the two
variables in increasing through time. Hence
there is unlikely to be a relationship.
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Cointegration in Single Equations

Example Yt 2 Yt-1 ut and Xt 1
0.5Xt-1 ut Yt I(1) Xt I(0)

Yt 2 Yt-1 ut
Xt 1 0.5Xt-1 ut
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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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Cointegration Main Conclusions
Estimated relationship between two independent
I(1) variables will typically be spurious. If
two I(1) variables cointegrate then there is a
long run relationship between the
variables. The residual regressions will be
I(0) i.e. do not have to be differenced to
produce stationary series.
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Cointegration Main Conclusions
Consequently to test for cointegration between
variables, we consider whether the residuals are
stationary from an OLS regression with the
variables. Test for cointegration using
informal and formal methods (as we did while
testing for unit root) (1) Plot regression
residuals and use correlogram or residual series
(2) Use Dickey Fuller tests on regression
residuals