Estimated coefficients structural VAR - svar

[Francois Durant]

Hi,

I am trying to estimate the coefficients of an svar model with exogenous variables. I understand how to impose restrictions and estimate the matrix A and B in Stata. However, I do not know how to output the estimated coefficients and relevant statistical significance in a way they can be interpreted.

In other words, I get the estimation of the unrestricted coefficients A and B (in matrix form):
A Y(t) = A0 + B Y(t-1) + C X(t) + u(t) (1)
with X exogenous variable and Y endogenous variable

but I would like to rearrange the VAR equation in a way that I can output the estimated coefficients and statistical significance as:
Y(t) = (A^-1)A0 + (A^-1)B Y(t-1) + (A^-1) X(t) + (A^-1) u(t) (2)

Here what I have and the estimate the coefficients from (1) from what I understand (although I did not include exogenous variable in this code):

webuse lutkepohl2
tsset
matrix A = (1, 0, 0, 0, 0 \ ., 1, 0, 0,  0 \ ., ., 1, 0,  0 \ ., ., ., 1,  0 \ ., ., ., .,  1)
matrix list A
matrix B = (., 0, 0, 0, 0 \ 0, ., 0, 0, 0 \ 0, 0, ., 0, 0 \  0, 0, 0, ., 0 \ 0, 0, 0 , 0, 1)
matrix list B
svar D.ln_inv D.ln_inc D.ln_consump consump inc if tin(,2005q4), aeq(A) beq(B) nolog lags(1)

I would like to display the coefficient and statistical significance in (2). Please help. Thanks!

[Eric de Souza]

Let B0 = (A^-1)A0 and B1 = (A^-1)B and B2 = (A^-1) .
B0, B1, B2 are highly non-linear in the A(ij) coefficients. One could calculate standard errors of each of the elements of B0, B1 and B2 using the delta method, but because of the high degree of nonlinearity, the estmates of the standard errors would be very unreliable.

[Francois Durant]

Thanks for the suggestion Eric, but it does not solve my problem since not Stata implementation is provided.

Would you be able to show me how to calculate the coefficients and the statistical significance (even if non reliable) in Stata ? Thanks.

[Eric de Souza]

You would have to write down the restrictions implied by the structural VAR on the reduced, impose those restrictions and estimate the reduced form subject to those restrictions. It would take too much time

[Francois Durant]

How do people interpret the structured VAR coefficient usually then? I understand that people often focus on Granger causality, IRF and variance decomposition.

That said, structured VAR is a sequence of OLS estimations. Coefficients estimated have a meaning if the variable considered in the model are stationary. This is a problem that isn't uncommon. Is there any ways to do that with Stata?

[Eric de Souza]

1.People interpret the model by looking at the impulse response functions.
2.If the variables are non-stationary you can always start with a Vector Error Correction Model and impose a structure on it, just as you would do with a VAR.

[Francois Durant]

I am not seeking to use alternative methods, I would like to find out how to estimate the coefficients I mentioned. Please let me know if you know how to do it. Thanks.