## Short-and-long run restrictions with VECM

Questions and discussions on Vector Autoregressions

### Short-and-long run restrictions with VECM

The following is an example of a structural VECM using short and long-run restrictions. It shows how to compute the sum of the moving average coefficients for a VECM, and impose the restrictions using the ShortAndLong procedure. Note that because the long-run response matrix is singular, some zero restrictions will be forced due to other zero restrictions; in this case, because the MASUMS matrix is rank one, any zero anywhere in a column will force all three variables to have a zero long-run response to that shock.

Code: Select all
`** Lutkepohl, New Introduction to Multiple Time Series Analysis* Example 9.4 from pp 377-383*open data kpswdata.ratcalendar(q) 1947data(format=rats) 1947:1 1988:4 c in y mp dp r** Data are already in logs. We need to scale by 100.*set c  = c*100.0set y  = y*100.0set in = in*100.0*graph(footer="Figure 9.1 Quarterly U.S. data",\$  key=below,klabels=||"Income","Consumption","Investment"||) 3# y# c# in** With two cointegrating vectors, this would be easier to analyze using* CATS.*@johmle(lags=2,det=constant,vectors=eigenvect)# y c in** Normalize first two eigenvectors to be identity in top 2 x 2 corner.*compute beta=%xsubmat(eigenvect,1,3,1,2)*inv(%xsubmat(eigenvect,1,2,1,2))** Define the two cointegrating relations for input into ECT*dec vect[equation] ecteqns(2)equation(coeffs=%xcol(beta,1)) ecteqns(1)# y c inequation(coeffs=%xcol(beta,2)) ecteqns(2)# y c in** Define and estimate the VECM*system(model=vecm)variables y c indet constantlags 1 2ect ecteqnsend(system)*estimate** Compute the long-run response matrix for the VECM. For a model* estimated with an ECT component, %varlagsums is the sum of lags on the* differences only.*compute masums=%perp(beta)*\$  inv(tr(%perp(%vecmalpha))*%varlagsums*%perp(beta))*\$             tr(%perp(%vecmalpha))*dec rect lr(3,3) sr(3,3)** Because of rank 2 cointegration, the masums matrix is rank one. As a* result, restricting the (1,2) and (1,3) long-run responses to zero* also forces the responses of the other two variables to zero as well.* So three restrictions (in this pattern) will just identify the model.*input lr . 0 0 . . . . . .input sr . . . . . 0 . . .@ShortAndLong(lr=lr,sr=sr,masum=masums) %sigma fdisp ###.### "Impact Responses" fdisp ###.### "Long run Responses" masums*f*@varirf(factor=f,steps=30,model=vecm,shocklabels=||"Permanent","Trans 1","Trans 2"||)`

Data file:

kpswdata.rat
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

dear
how to determine the permanent shock and graph the common trend? through weakly exogenous test?
thank you
luxu1983

Posts: 67
Joined: Wed Aug 12, 2009 10:53 pm

### Re: Short-and-long run restrictions with VECM

To get the permanent shock, use @StructResids (see http://www.estima.com/forum/viewtopic.php?f=4&t=684) and take the first component. The common trend is just the accumulation of that.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

TomDoan wrote:To get the permanent shock, use @StructResids (see http://www.estima.com/forum/viewtopic.php?f=4&t=684) and take the first component. The common trend is just the accumulation of that.

code like this ?
"@structresids(factor=f) resids %regstart() %regend() sresids"
is the first component sresids(1)?
luxu1983

Posts: 67
Joined: Wed Aug 12, 2009 10:53 pm

### Re: Short-and-long run restrictions with VECM

x and y are cointegration
how can i determine the who permanete shock ooriginates from ?
thank you very much
luxu1983

Posts: 67
Joined: Wed Aug 12, 2009 10:53 pm

### Re: Short-and-long run restrictions with VECM

luxu1983 wrote:
TomDoan wrote:To get the permanent shock, use @StructResids (see http://www.estima.com/forum/viewtopic.php?f=4&t=684) and take the first component. The common trend is just the accumulation of that.

code like this ?
"@structresids(factor=f) resids %regstart() %regend() sresids"
is the first component sresids(1)?

That's correct. You also have to add a RESIDS=resids option to the ESTIMATE.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

luxu1983 wrote:x and y are cointegration
how can i determine the who permanete shock ooriginates from ?
thank you very much

In general, you can't.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

TomDoan wrote:
luxu1983 wrote:x and y are cointegration
how can i determine the who permanete shock originates from ?
thank you very much

In general, you can't.

two variables need one restriction to identify, i do not do short restriction
i have no reason that who generate the permanete shock

so i let the permanete shock originates from the x or y respectively to get the different IRF

does this make sense?
luxu1983

Posts: 67
Joined: Wed Aug 12, 2009 10:53 pm

### Re: Short-and-long run restrictions with VECM

I would suggest you get a copy of The Cointegrated VAR Model by Juselius. It not only explains the mechanics of cointegration, but also how you give an economic interpretation to the results. If currencies A and B are out of PPP---is A high; is B low; both? The policy makers in country A would probably say B is low; the policy makers in country B would probably say A is high. The data aren't really equipped to answer that; that's why economists write papers with policy histories and discussions.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

Hi Tom, I am trying to estimate a SVECM and see how one of the variables behaves after imposing different long-run and short-run restrictions. I wonder if the "Impact Responses" and "Long run Responses" output from @ShortAndLong are the estimated coefficients of the SVECM, and if I can use them to obtain the 'fitted' values. What I have in mind is to use this coefficients, the permanent shocks from @StructResids and the initial values of the variable to get a Beveridge-Nelson MA representation of the SVECM. Is this a right procedure? and if not, how can I get the fitted values?
dacanoo

Posts: 6
Joined: Tue Aug 30, 2011 7:27 pm

### Re: Short-and-long run restrictions with VECM

dacanoo wrote:Hi Tom, I am trying to estimate a SVECM and see how one of the variables behaves after imposing different long-run and short-run restrictions. I wonder if the "Impact Responses" and "Long run Responses" output from @ShortAndLong are the estimated coefficients of the SVECM, and if I can use them to obtain the 'fitted' values. What I have in mind is to use this coefficients, the permanent shocks from @StructResids and the initial values of the variable to get a Beveridge-Nelson MA representation of the SVECM. Is this a right procedure? and if not, how can I get the fitted values?

The stochastic trend representation of the series of a VECM doesn't depend upon the structural model for the covariance matrix---it's a function of the lag parameters of the VECM and the non-orthogonalized innovations. If you impose a structural model on the innovation, you simply rotate the component trends and counter-rotate the weights to get the same result.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

Thank you Tom, I think I don't really understand the reason to impose short and long run restrictions in a VECM. I've seen in Johansen (1995) that cointegration coefficients can not be interpreted as elasticities, neither cointegrating relations as equations, but that a in SVECM it is possible. What I want is to have a right interpretation of long run cointegrating relations and see how changes in one of them affect the others. I thought this could be done with short and long run restrictions to obtain a SVECM, but if the stochastinc trends can not be associated to any particular series I can not interpretate them neither. What do you think I could do?
dacanoo

Posts: 6
Joined: Tue Aug 30, 2011 7:27 pm

### Re: Short-and-long run restrictions with VECM

If you have more than one stochastic trend, you can rotate the space to get trends behaving differently. It's just that the B-N type trend in any given series will be the same regardless of the SVAR that you apply.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

### Re: Short-and-long run restrictions with VECM

Dear Tom,
I am new to rats, can you clarify on a couple of questions I have? Thank you in advance.
1. This is regarding the long-run restrictions. Even if one wants the (two) transitory shocks to have no permanent effects, it suffices to restrict one row to zero (because of the reduced rank of the long run impact matrix). Is that a right statement to make? If so, does it matter which row I choose to put the zeros? Does this condition also holds if we have more than one common trends. For example, in the case two common trends (one cointegration relation), I have to put zeros on two rows for the transitory shock? Of course, in that case I would need one more long-run restriction and no short run restriction.
2. Is it possible to specify restricted trend in the cointegrating relations in johmle?
Cheers,
wfirew

Posts: 6
Joined: Thu Jun 21, 2012 8:16 am

### Re: Short-and-long run restrictions with VECM

wfirew wrote:Dear Tom,
I am new to rats, can you clarify on a couple of questions I have? Thank you in advance.
1. This is regarding the long-run restrictions. Even if one wants the (two) transitory shocks to have no permanent effects, it suffices to restrict one row to zero (because of the reduced rank of the long run impact matrix). Is that a right statement to make? If so, does it matter which row I choose to put the zeros?

Yes. As the model above is set up, any zero in any row of a column causes that shock for that column to have a zero LR effect on all variables.

wfirew wrote:Does this condition also holds if we have more than one common trends. For example, in the case two common trends (one cointegration relation), I have to put zeros on two rows for the transitory shock? Of course, in that case I would need one more long-run restriction and no short run restriction.

You can't have two shocks which have zero long-run effects on all variables in a model like that. A pair of zeros in a column will make that shock transitory on all three variables. Your final identifying assumption will need to be either one zero impact restriction or one zero long run restriction on one of the other two shocks.

wfirew wrote:2. Is it possible to specify restricted trend in the cointegrating relations in johmle?

It's not in the comments, but DET=RTREND will do the restricted trend.
TomDoan

Posts: 2720
Joined: Wed Nov 01, 2006 5:36 pm

Next