Is it possible to triangularize the VAR system in order to orthogonalize the innovations?

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Richard Mertens on 13 May 2016
Commented: Richard Mertens on 17 May 2016
Dear community,
I have a question about the estimation of Vector Autoregressive Models (VAR) using the Econometrics Toolbox. Is there any way to triangularize the VAR system in order to orthogonalize the innovations? As I understand it, is also called a recursive system. Any hints are greatly appreciated. Thanks in advance!

Hang Qian on 14 May 2016
Hi Richard,
I think it depends on the VAR specification.
If the triangularized model looks like
Y(t) = A1 * Y(t-1) + A2 * Y(t-2) + B * e(t),
where B is lower triangular and e(t) is N(0,1) noise without correlation,
then we can call the function VGXVARX to estimate a reduced-form VAR(2) model. The output EstSpec has a field EstSpec.Q, which is the estimated covariance matrix. Since B*B’ = Q, by cholesky decomposition chol(EstSpec.Q,’lower’), the loading matrix B is identified and recovered.
However, if the structural model is not exactly identified, say we’d like some contemporaneous matrix A0 such that
A0*Y(t) = A1 * Y(t-1) + A2 * Y(t-2) + e(t),
then the first step is still to estimate a reduced-form VAR(2) model using VGXVARX.
Y(t) = inv(A0)*A1 * Y(t-1) + inv(A0) * A2 * Y(t-2) + inv(A0) * e(t).
Since A0 contains n^2 elements, they cannot be individually recovered from data alone. It is necessary to impose identification restrictions. For example, we may triangularize the model by assuming that A0 is lower triangular. Since VGXVARX has reported inv(A0)*A1, inv(A0)*A2 and inv(A0) * inv(A0)’, we can recover the parameter by hand:
invA0 = chol(EstSpec.Q,’lower’);
A0 = inv(invA0 );
A1 = A0 * EstSpec.AR{1};
A2 = A0 * EstSpec.AR{2};
Of course, I used inv for illustration purposes. Backslash is the MATLAB style to do the matrix inversion.
- Hang Qian
Richard Mertens on 17 May 2016
Dear Hang, thanks a lot for your help. I am still not getting much further, though. I realize, I probably should have given more detail on my first question. What I need to do is estimate a VAR(1) model with k=5 variables. The first variable k=1 is a stock market return and I want the other four other error terms to be orthogonal to the error term of the stock market return. However, correlation between the remaining four variables shall be allowed. This is not achieved by diagonalization, correct? Is it possible to impose such restrictions? If yes, could you try to tell me how? Actually, this procedure is mentioned in the paper "Do the Fama-French Factors Proxy for Innovations in Predictive Variables" by Petkova (2006). However, I cannot figure out how it was done there. Again, thanks for any hints. Regards, Richard