VECM with some stationary variables

[bhpatterson]

I have a vector error-correction model with six endogenous variables: four are I(1) and two are I(0). Having used CATS to obtain cointegrating relations among the I(1) variables, I can "integrate" (sum) the two I(0) variables and then estimate the model in RATS. After I integrate, the code is as follows:

system(model=varmodel)
variables [6 endogenous variables]
lags 1 to 5
det constant
ect [error-correction terms]
end(system)
estimate(outsigma=s)

I can obtain impulse responses and variance decompositions as follows:

impulses(model=varmodel,steps=24)
errors(model=varmodel) * 24 s

However, the impulse responses and variance decompositions are given in levels rather than in differences. This is what I want for my four I(1) variables, but it poses a problem for my two integrated I(0) variables. Does anyone know of a way around this? Thanks very much for any advice.

[TomDoan]

A stationary variable is "cointegrated" with itself. If you add cointegrating terms which are empty equations like:

equation(empty) estat1 svariable1
equation(empty) estat2 svariable2

you'll get the proper behavior.

[bhpatterson]

Thank you for the tip. I use that code as follows. (Note: I named my cointegrating relations coint1 and coint2.)

equation(empty) estat1 dlcpi *(first stationary variable)
equation(empty) estat2 lunemp *(second stationary variable)

system(model=varmodel)
variables ldpc gs10 lpdi lcs
lags 1 to lags
det constant
ect coint1 coint2 estat1 estat2
end(system)
estimate(outsigma=s)


Below is the output for one of the equations in the VECM. Coefficients are not estimated for the two terms estat1 and estat2. Additionally, I am only able to use one lag. What should I change?

Dependent Variable LDPC
Quarterly Data From 1988:02 To 2009:04
Usable Observations 87 Degrees of Freedom 68
Mean of Dependent Variable 0.0081400111
Std Error of Dependent Variable 0.0094431385
Standard Error of Estimate 0.0065192543
Sum of Squared Residuals 0.0028900460
Durbin-Watson Statistic 1.896398

Variable Coeff Std Error T-Stat Signif
*******************************************************************************
1. D_LDPC(1) 0.126019179 0.125494186 1.00418 0.31885026
2. D_LDPC(2) -0.146589328 0.138535722 -1.05813 0.29373890
3. D_LDPC(3) 0.161658054 0.140805458 1.14810 0.25495226
4. D_LDPC(4) 0.036197758 0.133086710 0.27199 0.78645670
5. D_GS10(1) 0.002527870 0.002015680 1.25410 0.21409871
6. D_GS10(2) 0.001395772 0.001945089 0.71759 0.47546952
7. D_GS10(3) -0.000761653 0.001849172 -0.41189 0.68171655
8. D_GS10(4) 0.002534960 0.001704399 1.48730 0.14155713
9. D_LPDI(1) -0.060559257 0.108092146 -0.56026 0.57714612
10. D_LPDI(2) 0.191086507 0.109099535 1.75149 0.08437089
11. D_LPDI(3) -0.044781419 0.109911267 -0.40743 0.68497021
12. D_LPDI(4) -0.108509221 0.104047724 -1.04288 0.30069768
13. D_LCS(1) -0.080163746 0.060011081 -1.33582 0.18606322
14. D_LCS(2) 0.170883875 0.075309167 2.26910 0.02643936
15. D_LCS(3) -0.047262139 0.082689046 -0.57156 0.56950142
16. D_LCS(4) 0.024346664 0.077956770 0.31231 0.75576079
17. Constant 0.004366853 0.002327258 1.87639 0.06489313
18. EC1{1} -0.001194311 0.002074327 -0.57576 0.56667918
19. EC2{1} 0.090178340 0.033713270 2.67486 0.00935622
20. EC3{1} 0.000000000 0.000000000 0.00000 0.00000000
21. EC4{1} 0.000000000 0.000000000 0.00000 0.00000000

[TomDoan]

The two I(0) variables aren't in your model itself which is why they don't do anything. I was assuming that you had a mix of I(1) and I(0) variables in one model. Are you trying to make the I(0) variables exogenous as well?

[bhpatterson]

Sorry for the confusion. I do indeed have two I(0) variables (the unemployment and inflation rates) and four I(1) variables in a single model, all of which are supposed to be endogenous. If I include the I(0) variables on the line of code (shown in previous posts) beginning "variables...", they will enter into the VECM in differenced form, which is what I do not want. If I convert the I(0) variables into cumulative sums (so that they are now I(1)) and include them on the line of code beginning "variables...", the VECM will be estimated correctly.

My only problem is that the impulse responses are reported not in differences, but in levels. So for example, I am given the effect of a past shock to income on the current cumulative sum of the unemployment series, rather than on the current level of the unemployment series. And I am given the effect of a past shock to income on the current price level rather than on the current inflation rate. Perhaps I could difference the impulse responses for these two variables, but I wondered if there might be a better way.

[TomDoan]

Don't worry. Put them in the way that I said. It will work. When the model is substituted out (with %MODELSUBSTECT, they will be in levels like all the other variables, but won't have any unit root restrictions on their behavior.