Correct sign of the IRF

[BinhPham]

Hi Tom,

Is it valid if we switch the sign of two responses w.r.t a shock? For example, if I order (x1,x2,x3,x4) then I have the correct (expected) responses of (x1,x2,x3,x4) to an x4's shock according to Cholesky. Due to economic meaning, I instead order (x4,x1,x2,x3) then a shock to x4 gives the similar responses of x1 and x2 but with the reversed sign. That is x1 has positively initial response instead of negative as in the (x1,x2,x3,x4) order.

Is it correct if we simply flip the sign of x1 and x2 responses in the (x4,x1,x2,x3) case?

Many thanks your kind support,

[TomDoan]

If you do Cholesky in the order (x1,x2,x3,x4) then the impact on x1 (and x2 and x3) to a shock in x4 should be (by construction) zero. In the order (x4,x1,x2,x3), the signs of the impacts on x1, x2 and x3 will be the signs of their correlations with x4. You can flip the sign of a Cholesky shock if the own-impact has a sign different from what you would like (for instance, a negative shock to an interest rate would be "expansionary", if that's what you want to examine), but you should not try to compare one shock with a positive own-impact with another with a negative own-impact.

If you have a "theory", then you should probably be thinking about doing a structural model rather than Cholesky orderings.