SVAR model using CVMODEL

[kwangsooKo]

Dear Sir

I have three questions with regard to the SVAR model using CVMODEL as attached RDF file.

First, question is about identification. Using the three restrictions of c23=0, c32=0, and c13=-c12, the SVAR could be just-identified. Can I identify SVAR using not zero restriction on CVMODEL such as c13 =-c12
Second, found Log likelihood(2295.6133) is different from Log Likelihood Unrestricted(2345.0891). Is this identification problem? Please tell me what the cause of this and how to solve it?
--------------------------------------------------------------------------------------------------------------
NONLIN c12 c21 c31
dec frml[rect]c_form
frml c_form = ||1,c12,-c12|c21,1,0|c31,0,1||
com c12=0.01,c21=0.01,c31=0.01, c11=0.01, c22=0.01, c33=0.01
cvmodel(b=c_form,factor=c) %sigma
dis c
Covariance Model-Concentrated Likelihood - Estimation by BFGS
Convergence in 12 Iterations. Final criterion was 0.0000020 <= 0.0000100
Observations 209
Log Likelihood 2295.6133
Log Likelihood Unrestricted 2345.0891

Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. C12 4.8776145206 0.8219899079 5.93391 0.00000000
2. C21 0.0118883488 0.0088404324 1.34477 0.17869950
3. C31 0.0118498777 0.0102051766 1.16116 0.24557545
----------------------------------------------------------------------------------------------------------
Finally, I try to estimate SVAR model using CVMODEL in a different way as following. This result has no problem with Log likelihood. However, the restriction of c13=-c12 do not working well (C12 =0.00223 and C13= -0.0440, the sing is opposite but amount is not same). I know I do not standardized c_form matrix such that the diagonal elements are unity yield. To normalize unit shock, I restrict diagonal element as 1, it make problem with Log likelihood. So could you please tell me how to restrict on CVMOEL in this case?
----------------------------------------------------------------------------------------------------------
NONLIN c12 c21 c31 c11 c22 c33
dec frml[rect]c_form
frml c_form = ||c11,c12,-c12|c21,c22,0|c31,0,c33||
com c12=0.01,c21=0.01,c31=0.01, c11=0.01, c22=0.01, c33=0.01
cvmodel(b=c_form,factor=c) %sigma
dis c
Covariance Model-Concentrated Likelihood - Estimation by BFGS
Convergence in 21 Iterations. Final criterion was 0.0000057 <= 0.0000100
Observations 209
Log Likelihood 2345.0891
Log Likelihood Unrestricted 2345.0891

Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. C12 0.0052522049 0.0131592011 0.39913 0.68979893
2. C21 0.0151360619 0.0191753468 0.78935 0.42990741
3. C31 0.0191135050 0.0234042346 0.81667 0.41411780
4. C11 0.0733406143 0.1009318951 0.72663 0.46744978
5. C22 0.0085884155 0.0418135930 0.20540 0.83726142
6. C33 0.0002836078 0.0007352103 0.38575 0.69968143

0.01724 0.00223 -0.04400
0.00356 0.00364 3.35618e-018
0.00449 -1.37861e-019 0.00238

[TomDoan]

While your model has the parameter count right to just-identify the model, that pattern puts a (nonlinear) restriction on the set of covariance matrices which can be factored exactly. Since the empirical covariance matrix has probability 0 of hitting the restricted space, you won't be able to factor the matrix in practice. Again, while not apparent, you actually have only two free parameters in the model.

[kwangsooKo]

Thanks vary much for your answer. I need to think more carefully about this problem with covariance matrix.

I have a additional question about restriction on CVMODEL
I use the three restrictions of c23=0, c32=0, and c13=-c12 on CVMODEL
c13=-c12 means that sum of contemporaneous effect of third variable shock on second variable and contemporaneous effect of second variable shock on third variable are zero

The results are as following.
The restriction of c13=-c12 does not working well (C12 =0.00223 and C13= -0.0440, the sign is opposite but the absolute value is not same)
I want to restrict c13=-c12 on CVmodel. Please tell me how to solve it?
_________________________________________________________
NONLIN c12 c21 c31
dec frml[rect]c_form
frml c_form = ||1,c12,-c12|c21,1,0|c31,0,1||
com c12=0.01,c21=0.01,c31=0.01, c11=0.01, c22=0.01, c33=0.01
cvmodel(b=c_form,factor=c) %sigma
dis c

0.01724 0.00223 -0.04400
0.00356 0.00364 3.35618e-018
0.00449 -1.37861e-019 0.00238

[TomDoan]

True, but the variances of the second and third shocks aren't the same in that parameterization (with unit diagonals), hence the differences. If you want the third shock to have equal and opposite contributions from the unit size shocks, you need to include the diagonals in the parameter set:

NONLIN c11 c22 c33 c12 c21 c31
dec frml[rect]c_form
frml c_form = ||c11,c12,-c12|c21,c22,0|c31,0,c33||
com c12=0.01,c21=0.01,c31=0.01, c11=0.01, c22=0.01, c33=0.01
cvmodel(b=c_form,factor=c,dmatrix=identity) %sigma

[kwangsooKo]

Thanks very much for your kind reply
I used the CVmodel with diagonals in the parameter set.
Still, the problem of covariance matrix has been encountered, even I chang the endogenous variable.
The variance and covariance matrix of residuals seems to have hitting enough restricted space as followings.

Do I misread the problem with probability 0 of hitting the restricted space in covariance matrix?
So please tell me how to solve it excepting the change of identification form.
----------------------------------------------------------------------------------------------
NONLIN c12 c21 c31 c11 c22 c33
dec frml[rect]c_form
frml c_form = ||c11,c12,-c12|c21,c22,0|c31,0,c33||
com c12=0.02,c21=-1,c31=1, c11=1, c22=1, c33=1
cvmodel(b=c_form,factor=c,dmatrix=identity) %sigma
dis c
dis %sigma


Covariance Model-Likelihood - Estimation by BFGS
Convergence in 30 Iterations. Final criterion was 0.0000024 <= 0.0000100
Observations 208
Log Likelihood 476.9588
Log Likelihood Unrestricted 518.7283

Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. C12 0.022461831 0.004576259 4.90834 0.00000092
2. C21 -0.017706752 0.014905891 -1.18790 0.23487166
3. C31 -0.020887629 0.017461260 -1.19623 0.23160802
4. C11 -0.042383037 0.003598081 -11.77935 0.00000000
5. C22 0.178258834 0.008853639 20.13396 0.00000000
6. C33 0.188146043 0.009929054 18.94904 0.00000000

-0.04238 0.02246 -0.02246
-0.01771 0.17826 -0.00000
-0.02089 5.58531e-018 0.18815


0.00223
0.00233 0.03108
-0.00105 0.01963 0.03696
------------------------------------------------------------------------------------------------

[TomDoan]

Thanks very much for your kind reply
I used the CVmodel with diagonals in the parameter set.
Still, the problem of covariance matrix has been encountered, even I chang the endogenous variable.
The variance and covariance matrix of residuals seems to have hitting enough restricted space as followings.

Do I misread the problem with probability 0 of hitting the restricted space in covariance matrix?
So please tell me how to solve it excepting the change of identification form.

I can't. No one can. Not all (in fact, effectively no) covariance matrices can be exactly factored with your model. That's provable mathematically. If you don't believe me, take your original formulation with 1's on the diagonal, multiply that matrix by its transpose and write out the equations. You'll see that you have three equations in just two unknowns for the bottom 2x2 submatrix of the covariance matrix.

The problem isn't with the restriction on the top---it's with the bottom two rows. If you look at those in isolation:

?? ?? ??
x 0 x
0 x 0

the counting rule in RUBIO-RAMIREZ, J. F., D. F. WAGGONER, AND T. ZHA (2010): “Structural Vector Autoregressions: Theory of Identication and Algorithms for Inference,” Review of Economic Studies, 77(2), 665–696 says for a B factorization, to get a just identified model, we need one column with 1 non-zero, one with 2 non-zeros and one with 3 non-zeros. No matter what you do on that top row, you can't get a column with 3, which is what shows up if you expand the equations---there's not enough richness in the parameters for the bottom two rows (which determine, by themselves, the bottom 2x2 submatrix of the covariance matrix) to handle all covariance matrices.

[kwangsooKo]

Dear sir,
Your answers are being very helpful understanding SVAR.
Thank you very much for your great advice.