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### Cointegration analysis in practice

Engle-Granger 2 step, and 1 step, estimation procedures

Cointegration in Single Equations:
Cointegration -  evidence of long-run or equilibrium relationships
With cointegration the residuals from a regression are stationary.
Tested informally and formally for cointegration
Formal Tests include
(1) Cointegrating Regression Durbin Watson (CRDW) test
(2) Cointegrating Regression Dickey Fuller (CRDF) test

Summary of Lecture
(1) Introduce Granger Representation Theorem.
- relates cointegration to Error Correction Models
(2) Suggest different ways of estimating long run coefficients and short run models
(3) Multivariate regressions and testing for cointegration.

Cointegration: The usefulness of ECMs
Error correction mechanisms are useful for representing the short run relationships between variables.
Another way of saying we are not always at equilibrium.
The error correction model allows us to return to zero i.e. corrects for deviations from equilibrium.
It relates deviations from equilibrium to changes in the dependent variable i.e. the means of correcting for errors.

The estimation of two variable ECMs
However, are we certain an ECM relationship exists for variables? Does cointegration help?
yt = β0 + β1xt + ut
Granger Representation Theorem
Provided two time series are cointegrated, the short-term disequilibrium relationship between them can always be expressed in the error correction form.

Cointegration and ECMs
Granger Representation Theorem suggests that if we have cointegration then an ECM exists
Dyt = lagged (Dyt , Dxt) – lut-1 + et
ut-1              is the disequilibrium error yt = β0 + β1xt + ut
l             is the short-run adjustment parameter
This is an important result since it is justification for using ECM.
If yt and xt are cointegrated then the disequilibrium errors ut will be stationary.

This means there is a force pulling the residual errors towards zero.
Dyt = lagged (Dyt , Dxt) – lut-1 + et
Notice that all first differenced variables are I(0).
Disequilibrium errors (ut-1 ) also need to be I(0).
This is the case when yt and xt are cointegrated.
Exact lags are not specified by the Granger Representation Theorem. Specification determined by general to specific approach.
Since Dxt-1 is an I(0) variable so is Dxt
Hence it is possible to incorporate unlagged values of D xt
(but may then need to use Instrumental Variables regression).

Estimating ECMs using Cointegration
How do we obtain an error correction model?

Engle-Granger Two-Step approach
(1) Estimate long run relationship between yt and xt
(2) Incorporate residuals in a short run model

Engle-Granger Two-Step approach
(1) Estimate long run relationship between yt and xt
yt = β0 + β1xt + ut
- When there is cointegration we can be confident that  β0 and β1 will not be biased (in large samples).
As Stock suggested β0 and  β1 are consistent.
Also superconsistent. We can ignore dynamic terms.

Now use the residuals from the ‘cointegrating regression’ to test for cointegration (i.e. the existence of a long run equilibrium relationship)
Use the residuals of the estimated long run relationship, test (using DF/ADF statistics) whether or not u is STATIONARY

Note: must use special tabulated critical values for CRDF/CRADF tests.
If the residuals are stationary, then we can conclude that the series are COINTEGRATED

Engle-Granger Two-Step approach
(2) Incorporate residuals in a short run model
We take the residuals from the estimated static equation ut-1 and incorporate them into the short run model.
Dyt = lagged (Dyt , Dxt) – l ut-1 + et
We consequently estimate this regression.
We can do so by OLS since all the variables are stationary.
We should obtain the estimated coefficient l

Engle-Granger Two Step
Problems with the Engle-Granger Two Step
These are concentrated on the first step.
- estimating the static OLS model.
We suggested that OLS estimates of cointegrating regressions will be unbiased in large samples (consistent).
However there may be bias in small samples (the samples we use).
If there is bias in the first step, this will spillover on to the second step.
Typically residuals are only used to test cointegration
One suggestion is that long run parameters should be estimated using  methods unbiased in small samples, the implied residuals derived and then the short run model estimated.

Engle Granger Approach becomes
(1) Use AutoRegressive Distributed Lag (ARDL) method to estimate parameters
i.e. within a dynamic model
(2) Derive the residuals errors from the long run model
ut = yt - β0 - β1xt
(3) Incorporate residuals in the error correction model
Dyt = lagged (Dyt , Dxt) – l ut-1 + et
Alternative suggestion is that short run and long run parameters should be estimated in a single step to avoid bias estimates (in small samples).

Banerjee, Dolado, Hendry and Smith (1986) method
Dyt = lagged (Dyt , Dxt) – lut-1 + et
Dyt = lβ0  + lagged (Dyt , Dxt) – l yt-1 + l β1xt-1  + et
where   ut  = yt - β0 - β1xt
Simulation studies of the properties of this estimator, suggest that in small samples Banerjee et al. approach performs better than Engle-Granger method.

Banerjee, et al. (1986) approach
Dyt = lβ0  + lagged (Dyt , Dxt) – l yt-1 + l β1xt-1  + et
where   ut  = yt - β0 - β1xt
Can check cointegration by testing the residuals et for stationarity
Although there are two I(1) variables in this equation a linear combination should cointegrate to produce a stationary relationship.
Consequently all variables (or a combination of variables)  will be I(0) and inference can proceed as normal.

Multivariate Cointegration Tests

Johansen Approach
We have concentrated on the bivariate case yt and xt.
There can only be one cointegrating relationship between these variables.
Is this the case when there are three variables?
It may be the case that there is more than one relationship.
Where we have variables yt , xt and  zt.
Johansen approach not only examines if yt , xt and  zt cointegrated.
But also if yt cointegrates with xt on its own and yt cointegrates with zt  on its own.

Single Equation Approach
Dyt = lagged (Dyt , Dxt) – lut-1 + et
Soren Johansen Approach
Can test for the number of cointegrating relationships.
Assuming    yt cointegrates with xt                 = LR1
yt cointegrates with zt             = LR2
Short run model becomes
Dyt = lagged (Dyt Dxt Dzt ) – l 11LR1t-1 - l 12LR2t-1 + e1t
Dxt = lagged (Dyt Dxt Dzt ) – l 21LR1t-1 - l 22LR2t-1 + e2t
Dzt = lagged (Dyt Dxt Dzt ) – l 31LR1t-1 - l 32LR2t-1 + e3t

Testing for, and estimating, a cointegrating relationship
• Pretest the variables for the their order of integration
• Estimate the Cointegration Regression
• Check whether there is a cointegrating (i.e. long run equilibrium) relationship
• If so, estimate the dynamic error correction model