Finally, we shall take a look at this very elegant method introduced by Sargent. It consists of the following steps:
1. Write the model in terms of lead – and lag-polynomials in expectations.
2. Factor the polynomials, into one-order polynomials, deriving the roots.
3. Invert the factored one-order polynomials into the directions of converging forward polynomials of expectations.
Again, we use the simplifying definition
zt = bp2Xt + £pt:
so the model is again
Apt = bp1EtApt+i + bpiApt-i + zt.
Note that the forward, or lead, operator, F, and lag operator, L, only work on the variables and not expectations, so:
LEtzt = Etzt-i
FEtZt = Etzt+i L-i = F.
The model can then be written in terms of expectations as:
-bpiEtApt+i + EtApt – bpEApt-i = EtZt, and using the lead – and lag-operators:
(-bpiF + 1 — bpiL)EtApt = Etzt,
or, as a second-order polynomial in the lead operator:
The polynomial in brackets is exactly the same as the one in (A.12), so we know it can be factored into the roots (A.13):
However, we know that (1/1-(1/a2)F) = ^=0(1/a2)iFi, since |1/a2| < 1,
so we can write down the solution immediately:
where we have also substituted back for zt.
To derive the complete solution, we have to solve for
(1 bXL)xt Єxt.
We can now appeal to the results of Sargent (1987, p. 304) that work as follows. If the model can be written in the form
yt = Etyt+- + xta(L)xt + et,
a(L) =1 -^2 aj L°
with the partial solution
yt = (A)i Et’xt+i,
1 – XL-
r — 1
a(X)—1 1 + £ Y, Xk—jau I Lj
The solution therefore becomes Apt – a.1 Apt—1 = 1
then the complete solution