Triangular Factorization (non-orthonormal covariance matrix)

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short_run
Posts: 2
Joined: Thu Jan 03, 2019 3:15 am

Triangular Factorization (non-orthonormal covariance matrix)

Postby short_run » Fri Jan 04, 2019 2:23 am

Dear all,

I was wondering wether is it possible in Eviews to estimate a VAR and retrieve the structural impulse responses through the triangular factorization.

This is similar to the "Cholesky factorization" implemented in Eviews but the structural covariance matrix is diagonal with values which are not normalised to one as in the case of Cholesky factorization.

According to the manual the covariance matrix is assumed to be orthonormal in all cases, I couldn't find a way to relax this assumption.

Thank you!

EViews Matt
EViews Developer
Posts: 557
Joined: Thu Apr 25, 2013 7:48 pm

Re: Triangular Factorization (non-orthonormal covariance matrix)

Postby EViews Matt » Fri Jan 04, 2019 2:53 pm

Hello,

What you suggest could cause a change in the magnitude of the impulse responses, but not their overall shape. EViews' assumption that the structural innovations are orthonormal is reflected in the structural impulse response generation algorithm that uses unit (magnitude one) structural innovations. If you were to allow the structural innovation covariance matrix to be diagonal, rather than the identity matrix, what magnitude structural innovations would you want to use for generating the impulse responses?

short_run
Posts: 2
Joined: Thu Jan 03, 2019 3:15 am

Re: Triangular Factorization (non-orthonormal covariance matrix)

Postby short_run » Thu Jan 10, 2019 9:21 am

Dear Matt,

Thank you for your reply.

Yes, as for the IRFs it is just a matter of scaling but there are applications where the choice between Cholesky and triangular it is not neutral.

I guess the magnitude for the structural innovations (for generating the impulse responses) should be equal to the estimated standard devations, coming from the diagonal variance covariance matrix.

EViews Matt
EViews Developer
Posts: 557
Joined: Thu Apr 25, 2013 7:48 pm

Re: Triangular Factorization (non-orthonormal covariance matrix)

Postby EViews Matt » Thu Jan 10, 2019 11:15 am

Mathematically, it's not clear to me that there's any practical difference between the Cholesky and triangular factorization. Using what EViews calls an A-B SVAR model, i.e. A * y = <regressors> + B * u, you can certainly "pretend" that the structural innovations have diagonal covariance by restricting B to be diagonal (pretending B * u are the structural innovations rather than just u). Restricting A to be lower triangular (plus some additional restrictions), and assuming that the residual covariance matrix is properly positive-definite, A^-1 * B will still be equal the Cholesky decomposition and the IRFs will not change.


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