1. 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 isn’t 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.

2. Long-run Relationships
Consider the following static regression between two variables
Yt = β0 + β1Xt + ut
This relationship has the disequilibrium error ut (a linear combination of Yt and Xt) where:
ut =Yt - β0 - β1Xt
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”

3. Single Equations Errors+ ut -
Disequilibrium errors (i.e. ut = Yt - β0 - β1Xt)
No tendency to return to zero
Error rarely drifts from zero

4. 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

5. 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.

6. 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 > 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).

7. 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 = Xt-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.

8. Cointegration in Single Equations
Example
Yt = 2 + Yt-1 + ut and Xt = Xt-1 + ut
Yt ~ I(1) Xt ~ I(0)
Yt = 2 + Yt-1 + ut
Xt = Xt-1 + ut

9. 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 = θ0 + θ1Xt + θ2Yt-1 + θ3Xt-1 + ut
Static models estimated by OLS will be bias or inconsistent.
Yt = β0 + β1Xt + ut
Stock (1987) found that if Yt and Xt are cointegrated then OLS estimates of β0 and β1 will be consistent.

10. 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.
=> (i) simple static regression don’t necessarily give spurious results.
(ii) dynamic misspecification is not necessarily a problem.
Consequently we can estimate simple regression
Yt = β0 + β1Xt + ut
even if there are important dynamic terms
Yt = θ0 + θ1Xt + θ2Yt-1 + θ3Xt-1 + ut

11. 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.

12. 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.

13. 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