1. ECO 400 Vector Error Correction Models (VECM)
Prof. Didar Erdinç

2. Testing for Unit Roots
First we test to see if variables are stationary i.e. I(0). If not they are assumed to have a unit root and be I(1).
If a set of variables are all I(1) they should not be estimated using OLS, as there may be one or more long-run equilibrium relationships, i.e. cointegration. We can estimate how many “cointegrating vectors” exist among the variables using Johansen’s technique.

3. Johansen’s Test for Cointegration
If a set of variables are found to have one or more cointegrating vectors then a suitable estimation technique is a VECM (Vector Error Correction Model) which adjusts to both short run changes in variables and deviations from equilibrium.
Suppose we have data on monthly unemployment rates in Indiana, Illinois, Kentucky, and Missouri from January through December 2003.
We believe in accordance with theory that labor market “factor mobility” will keep the unemployment rates in a LR equilibrium. The following graph plots the data.

4. Stata Application- Urates in US States
. use clear or . webuse urates .line missouri indiana kentucky illinois t This last command generates the next plot. Note the form of the above line to draw the line graph; then the variables which will be plotted; finally t the time variable against which they are all plotted.

5. Plotting Urates

6. Graphs- Interpretation
The graph shows that although the series do appear to move together, the relationship is not smooth. There are periods when there are significant deviations from a long-run pattern of comovement in the series, i.e. short-run disequilibria. As an example, Illinois has the highest unemployment rates as compared to others in certain periods while Indiana has the lowest rate.
Although the Kentucky rate moves closely with the other series for most of the sample, there is a period in the mid-1980s when the unemployment rate in Kentucky does not fall at the same rate as the other series.
We will model the series with two cointegrating equations (after varsoc and vecrank commands, we find there are 2 cointegrating vectors and 4 lags based on AIC).
For now we use the noetable option to suppress displaying the short-run estimation table.

7. vec missouri indiana kentucky illinois, trend(rconstant) rank(2) lags(4) noetable

8. VEC cont.
Except for the coefficients on kentucky in the two cointegrating equations and the constant term in the first, all the parameters are significant at the 5% level.
We can refit the model – if we choose -with the Johansen normalization and the overidentifying constraint that the coefficient on kentucky in the second cointegrating equation is zero.
.constraint 1 [_ce1]missouri = 1
.constraint 2 [_ce1]indiana = 0
.constraint 3 [_ce2]missouri = 0
.constraint 4 [_ce2]indiana = 1
. constraint 5 [_ce2]kentucky = 0

9. . vec missouri indiana kentucky illinois, trend(rconstant) rank(2) lags(4) noetable bconstraints(1/5)
Note with these constraints, we are eliminating Kentucky from ce2 (constraint 5) while setting Missouri also equal to zero (as it is a small number).
Constraint number 1, [_ce1] tells us which equation and missouri=1 sets constraint.
Alternatively, we can drop Kentucky from the VEC estimation.

10. The result from the last VEC estimation with constraint and LR Test of identifying restriction
The test of the overidentifying restriction does not reject the null hypothesis that the restriction is valid, and the p-value on the coefficient on kentucky in the first cointegrating equation indicates that it is not significant. We could instead consider removing kentucky from the model.

11. Interpretation of ce1 and ce2
Next, we look at the estimates of the adjustment parameters. In the output below, we replay the previous results.
vec missouri indiana kentucky illinois, trend(rconstant) rank(2) lags(4) bconstraints(1/5)
Command generates the following output with the actual VEC results displayed.

12. Results for D_Missouri

13. Interpretation: CE1 (a negative coef.)
If the error term in the first cointegration relation (ce1) is positive unemployment in Missouri DECREASES (AS THE SIGN IS NEGATIVE AND SIGNIFICANT).
If the error term in the second cointegrating regression (ce2) is positive then unemployment in Missouri INCREASES.
The first cointegrating regression is Missouri Kentucky – 1.037Illinois = Error
if the error term is positive then unemployment in Missouri can be viewed as being above equilibrium, same for Kentucky, but for Illinois it is below equilibrium (because if we increase Illinois the error term falls)
To get back to equilibrium we need unemployment to fall in Missouri: - CE1

14. Interpretation: CE1 (a negative coef.)
As we can see from the regression this is what we get.
D_missouri is the change in unemployment in Missouri i.e. DUMt = Umt – Umt-1
The coefficient on _ce1 L1 (_ce1 : the error term from the first cointegrating regression; L1 lagged one period) is and significant at the 1% level.
This confirms our expectation that there is adjustment towards LR equilibrium. Thus if in period t-1 the error term in _ce1 was positive, which we can see can be seen as unemployment in Missouri being too high compared to the equilibrium relationship with the other two states, then d_missouri must fall and this can happen if coefficient for ce1 is negative and significant. The bigger the (negative) coefficient on _ce1 L1 the more rapid is the correction. If it is equal to -1 then the entire error is corrected for in the following period.

15. Interpretation: CE2 (a positive coef.)
Note the second cointegration vector (ce2) can be written as: Error=Indiana Illinois
From the vec output, we can see that the coefficient of ce2 in d_missouri equation is positive and significant which means Unemployment in Missouri increases if this is error term is positive. But why? Missouri does not enter the second cointegrating vector. So why does unemployment in it respond to it? See below for the output.

16. Interpretation: CE2 (a positive coef.)
Ce2Error=Indiana Illinois
This is a little convoluted, but if the error term is positive it suggests that unemployment in Illinois is below equilibrium (or Indiana above equilibrium) (and illinois may increaseor Indiana may decrease as a consequence).
Now from first cointegrating vector:
Missouri Kentucky – 1.037Illinois = Error or
Missouri = Kentucky Illinois, if Illinois unemployment is to increase then the error term in the first cointegrating vector will fall (perhaps going negative) so to make it zero again, Missouri should increase, i.e. d_missouri should be positively related to ce2 (L2) error (if positive error in ce2, then d_missouri should adjust positively giving a coefficient on ce2 in d_missouri equation which is positive! And significiant).

17. Results for D_Indiana
The error term from _ce1 is not significant, but that from _ce2 is and it is positive. _ce2 is lagged Error=Indiana Illinois
Now this does not make much sense if error term is positive, then unemployment in Indiana needs to fall to restore equilibrium. Yet the coefficient on it is positive indicating the opposite.
TO BE CONTINUED…