VAR inversion - looking for a good resource

Question

I am having trouble with something that should be pretty basic.

I need to invert a VAR (vector autoregression). Everything I have read just brushes past the actual inversion process, taking for granted that I already know how to invert a VAR. Does anyone know a good source (book, journal article, website) that walks through inverting? I've looked at Diebold's Elements of Forecasting and Enders' Applied Economic Times Series, as well as other sources with good coverage of VARs, but I can't find a good explanation of the basics of working with VARs.

Thanks!

Answer 1

To get Quah and Vahey's method, you should probably get a good handle on the Wold Representation Theorem first. Christiano has good notes on it here. Then, you can apply it to VAR. There's a pretty good basic explanation in Greene's Econometric Analysis. In the Sixth Edition, it's in Chapter 20 (Models with Lagged Variables) starting with Section 20.6, "Vector Autoregressions." It should be pretty straight forward to apply the Wold Theorem to VARs with Greene's notation. If it's still not quite clear to you, there's a pretty close example on page 704, subsection 20.6.8.b, "The Sacrifice Ratio." The basic idea is that you just end up sort of inverting the matrix of lag operators and coefficients (the "inversion" part) so that you have your main variable on the left side and a lag polynomial in your errors on the right side.