Recovering Structural Shocks in Long-run restricted SVARS

[BrsG]

I have estimated a 3-variable SVAR (EViews 11, latest patch), with long-run restrictions on F in

Psi*e = Fu

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_y.append(svar) @VEC(F) = NA, NA, NA, 0, NA, NA, 0, 0, NA

For the following note that u and e are related by Su = e, where S = A^(-1)B. Moreover, the underlying VAR is stable. The notation in this post tries to approximate the one used in the EViews documentation.

When trying to recover the orthonormal structural shocks u I ran into trouble and have the following issues.

(1) the residuals S^(-1) e, which should give us the structural shocks, are far from orthonormal. In particular, they are highly cross-correlated.

(2) Psi S != F, although identity is clearly implied (I calculated Psi = (I_3- A1-A2)^(-1) )

(3) Although the general approach with restrictions on F should not allow to get estimates for A or B, both are in fact available when calling

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matrix bmat = _y.@svarbmat matrix amat = _y.@svaramat

A turns out to be the identity matrix (and B=S). It seems the estimation procedure makes this assumption on A automatically whenever restrictions are placed on F. Even when trying prevent restrictions on A via

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_y.append(svar) @VEC(F) = NA, NA, NA, 0, NA, NA, 0, 0, NA _y.append(svar) @VEC(A) =NA, NA, NA, NA, NA, NA, NA, NA, NA

A is still the identity matrix.

What am I missing here?

Any help would be much appreciated!

[EViews Matt]

Hello,

Regarding (1) and (2), here's a quick program that verifies that the S and F matrices behave as expected for a trivial artificial SVAR. If you run the last four statements for your own SVAR, do the results not match the mentioned expectations?

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create u 100 series x = rnd series y = 2 * x + nrnd series z = nrnd var v.ls 1 2 x y z v.svar(preset=3) ' F is lower triangular. v.makeresids(struct, method=struct) x_r y_r z_r cov x_r y_r z_r ' Should be close to the identity matrix. matrix psi_s_minus_f = @inverse(@identity(3) - v.@lagcoefsum) * v.@svarsmat - v.@svarfmat show psi_s_minus_f ' Should be close to all zeros.

Regarding (3), you're correct that restricting S and/or F doesn't permit estimating A or B. For historical reasons, @svaramat and @svarbmat need to be available and meaningful after an estimation, so A is simply set to I and B is set to S. These aren't independent estimations, they're just one possible choice for A and B this is consistent with the estimation results.

[BrsG]

Thanks, Matt, this was helpful, especially on 3), which in combination with the others created the confusion.

I should have clarified in my question 1) that I had checked

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v.makeresids(struct, method=struct) x_r y_r z_r cov x_r y_r z_r ' Should be close to the identity matrix.

and it worked. However I also checked S^(-1)e = u, whose covariance matrix was off. I now found that the reason behind this was that I had created the residual matrix e via a group command with a wildcard, which had changed the ordering of the series.

So all good now. Thanks again!