Sign restrictions with non-recursive decomposition

[AmateurRatsUser]

Hey Tom,

I have been trying to calculate impulse responses of a 7 variable system with sign restrictions. I am really confused regarding how to proceed forward with the estimation.

Q1 : Say I estimate a standard SVAR using contemporaneous restrictions (just identified model). However I want to identify just two shocks based on sign restrictions (tax and expenditure) like in Dungey and Fry (2011), Economic modelling. How should I proceed?

Q2 : What's bothering me is that under sign restrictions, impulse responses are generated assuming recursive decomposition. Don't you think the identification under sign restrictions would be much better if supplemented by non-recursive decomposition supported by some theory? If yes, then how does one go about it?

Q3: Say if I want to get impulse responses of all the variables for all the shocks, do I need to identify all the shocks one by one like Mountford and Uhlig (2009) do for 4 shocks?

Looking forward to your response.

[TomDoan]

Hey Tom,

I have been trying to calculate impulse responses of a 7 variable system with sign restrictions. I am really confused regarding how to proceed forward with the estimation.

Q1 : Say I estimate a standard SVAR using contemporaneous restrictions (just identified model). However I want to identify just two shocks based on sign restrictions (tax and expenditure) like in Dungey and Fry (2011), Economic modelling. How should I proceed?

Don't bother with the SVAR. Just do the sign restrictions based on the original VAR.

Q2 : What's bothering me is that under sign restrictions, impulse responses are generated assuming recursive decomposition. Don't you think the identification under sign restrictions would be much better if supplemented by non-recursive decomposition supported by some theory? If yes, then how does one go about it?

The factorization is irrelevant, so using the Cholesky is simplest. See the original Uhlig paper for the theory about "impulse vectors".

Q3: Say if I want to get impulse responses of all the variables for all the shocks, do I need to identify all the shocks one by one like Mountford and Uhlig (2009) do for 4 shocks?

If you don't have a theory for the 3rd and 4th shocks, then don't try to figure one out just to fill out a decomposition. If you have a theory for the remaining shocks, then yes. There are ways of generating the whole batch then rejecting the full set if they don't meet the criteria, but they're equivalent to the one-at-a-time identification.

[AmateurRatsUser]

Hey Tom,

Thank you so much for the prompt reply.

AmateurRatsUser wrote:Hey Tom,

Q1 : Say I estimate a standard SVAR using contemporaneous restrictions (just identified model). However I want to identify just two shocks based on sign restrictions (tax and expenditure) like in Dungey and Fry (2011), Economic modelling. How should I proceed?

A: Don't bother with the SVAR. Just do the sign restrictions based on the original VAR.

I have one follow up question. What I meant by Q1 is - say I want to impose sign restrictions on only two shocks to identify them. However, the rest (5) I want to identify using the standard short-run restrictions because I am not sure of their sign behavior. Am I wrong in conceptualizing such an identification? How do I go about it? I want to generate impulse responses of every shock on every variable (possibly with different identification techniques).

[TomDoan]

In a structural VAR "B" model, the two impulse vectors generated by using sign restrictions will be the first two columns in the B matrix. You would treat those as known. The remaining five columns could be constructed as with any B model SVAR, except that the 6 and 5 restriction columns have been taken by the two impulse vectors. So you would need columns with 4, 3, 2, 1 and 0 restrictions. If you come up with that, you can estimate that using CVMODEL. Of course, you would have to re-estimate that with each accepted draw for the sign restrictions, so it might take a while to do all the number crunching.

[AmateurRatsUser]

Thanks Tom.

The proposed identification is much clearer to me now . I'll start coding it, and post it once the entire code is ready.

But a few more questions pop-up in my mind while visualizing such an identification:

1. The identification would always have sign restrictions first and short-run restrictions second. I am presuming that the other way around it is not possible i.e short-run identification first and sign restrictions later?

2.

In a structural VAR "B" model, the two impulse vectors generated by using sign restrictions will be the first two columns in the B matrix. You would treat those as known. The remaining five columns could be constructed as with any B model SVAR, except that the 6 and 5 restriction columns have been taken by the two impulse vectors. So you would need columns with 4, 3, 2, 1 and 0 restrictions. If you come up with that, you can estimate that using CVMODEL. Of course, you would have to re-estimate that with each accepted draw for the sign restrictions, so it might take a while to do all the number crunching.

If I understand correctly, once 2 shocks are identified using sign-restrictions we are left with 5 unidentified shocks. "B" matrix which is 7*7 requires restrictions on the sub-set of "B" (5*5) with 10 restrictions (because first two columns are filled using sign restrictions approach). You have a kind of cholesky decomposition, is it possible to go for a non-recursive decomposition here? (my original forum question)

[TomDoan]

Thanks Tom.

The proposed identification is much clearer to me now . I'll start coding it, and post it once the entire code is ready.

But a few more questions pop-up in my mind while visualizing such an identification:

1. The identification would always have sign restrictions first and short-run restrictions second. I am presuming that the other way around it is not possible i.e short-run identification first and sign restrictions later?

The only things that could go first are trivial constructive shocks like the start of a Cholesky factorization.

2.

In a structural VAR "B" model, the two impulse vectors generated by using sign restrictions will be the first two columns in the B matrix. You would treat those as known. The remaining five columns could be constructed as with any B model SVAR, except that the 6 and 5 restriction columns have been taken by the two impulse vectors. So you would need columns with 4, 3, 2, 1 and 0 restrictions. If you come up with that, you can estimate that using CVMODEL. Of course, you would have to re-estimate that with each accepted draw for the sign restrictions, so it might take a while to do all the number crunching.

If I understand correctly, once 2 shocks are identified using sign-restrictions we are left with 5 unidentified shocks. "B" matrix which is 7*7 requires restrictions on the sub-set of "B" (5*5) with 10 restrictions (because first two columns are filled using sign restrictions approach). You have a kind of cholesky decomposition, is it possible to go for a non-recursive decomposition here? (my original forum question)

Yes. That was what I was describing. You need the create a B matrix for CVMODEL with two fixed columns and free parameters in the remaining five.

[AmateurRatsUser]

Hey Tom,

Thank you for clarifying my doubts. I have managed to code the above problem (hopefully).

I have a few questions about the code (see the attached file):

1. I want to identify shock 5,6,7 through sign restrictions and rest through standard short-run restrictions, accordingly I have set up a CVMODEL in the loop. Could you please have a look and tell me if this is the right way to do it before I start interpreting the results?

2. When I construct impulse responses in the end using MCGRAPHIRF, I get reasonable IRFs for 5,6,7 shocks however other shocks remain close to zero? Can't understand why it is happening?

Looking forward to your response.

[TomDoan]

You have a few problems with that. First, your first and third columns are identical other than normalization, so they won't be identified. Also, you are violating the global identification criteria from Rubio-Ramirez--Waggonner and Zha(2010)---your four columns have to have (in some order) 3, 2, 1 and 0 zeros. Instead, you have 0,3,0,3. You have the correct number of restrictions, but not the correct pattern.

Finally, the sign-restricted columns are normalized to shocks of unit variance, and you can't combine those with columns that are normalized with unit loadings. Instead, you have to use DMATRIX=IDENTITY and replace the 1's in your SVAR with free parameters so that everything is normalized the same way.

[AmateurRatsUser]

Hey Tom,

Thanks for all the help.

I have altered the code as per your suggestions. But the issue still persists.

2. When I construct impulse responses in the end using MCGRAPHIRF, I get reasonable IRFs for 5,6,7 shocks however other shocks remain close to zero? Can't understand why it is happening?

Am I missing out on something?

Note: I have further modified the code to restrict the impact of sign restricted shocks on certain variables to be zero.

I would be grateful if you could tell me where I am going wrong.

[TomDoan]

Have you tried doing just a few draws, and putting the PRINT back on the CVMODEL to make sure the results of that look OK before doing the full Monte Carlo? You have an obvious problem that your guess values for the CVMODEL are fully zeroing out the first four columns. I'm not sure how that's going to work.

Also, does "small" mean small in an absolute sense, or small relative to their error bands. I wouldn't be surprised if the error bands on the first four are quite wide since they are what's left over after the three sign restricted shocks.

[AmateurRatsUser]

Hey Tom,

Actually, I get median impulse responses to be close to zero (almost flat) and very wide error bands for shocks identified using zero restrictions.

Does this mean that this identification scheme of combining two strategies does not work?

Incidentally, Dungey/Fry (2009) , Bjørnland/Halvorsen (2008) and Jaroci´nski/Smets (2008) etc. all use recursive decomposition with sign restrictions. As per my knowledge, I haven't seen any paper which combines non-recursive decomposition with sign restrictions.

[TomDoan]

You'll have to post the full program and the data set so we can take a look. It's quite possible that the sign restrictions leave such a varying degree of remaining correlation that the structural model is very imprecisely estimated.