860 Macroeconometric Modeling for Policy
Lettingyt≡(rw−βz)tandxt≡ztthen gives:
rwt=−ac+a(rw−βz)t− 1 +
a
2
%(rw−βz)t− 1 +βzt−ab%zt− 1 −
ab
2^2 zt− 1 +···.17.2.4 System representations
The approach easily generalizes to a system representation. For ease of exposition,
we illustrate the two-dimensional case for whichy 1 →y ̄ 1 andy 2 →y ̄ 2 ast→∞.
Expanding with respect toy 1 andy 2 about their steady-state values yields:
⎡
⎢⎣
f 1(
y 1 ,y 2)f 2(
y 1 ,y 2)⎤
⎥⎦=⎡
⎢⎣f 1(
y ̄ 1 ,y ̄ 2)f 2(
y ̄ 1 ,y ̄ 2)⎤
⎥⎦+⎡
⎢⎢
⎣∂f 1 (y ̄ 1 ,y ̄ 2 )
∂y 1∂f 1 (y ̄ 1 ,y ̄ 2 )
∂y 2
∂f 2 (y ̄ 1 ,y ̄ 2 )
∂y 1∂f 2 (y ̄ 1 ,y ̄ 2 )
∂y 2⎤
⎥⎥
⎦⎡
⎢⎣y 1 −y ̄ 1y 2 −y ̄ 2⎤
⎥⎦+⎡
⎢⎣R 1R 2⎤
⎥⎦,where[R 1 ,R 2 ]′denotes the vector:
1
2!⎡
⎢⎢
⎢⎢
⎣∂^2 f 1 (ζ,η)
∂y^21(
y 1 −y ̄ 1) 2
+ 2
∂^2 f 1 (ζ,η)
∂y 1 ∂y 2(
y 1 −y ̄ 1)(
y 2 −y ̄ 2)
+
∂^2 f 1 (ζ,η)
∂y^22(
y 2 −y ̄ 2) 2∂^2 f 2 (ζ,η)
∂y^21(
y 1 −y ̄ 1) 2
+ 2
∂^2 f 2 (ζ,η)
∂y 1 ∂y 2(
y 1 −y ̄ 1)(
y 2 −y ̄ 2)
+
∂^2 f 2 (ζ,η)
∂y 22(
y 2 −y ̄ 2) 2⎤
⎥⎥
⎥⎥
⎦,so that: ⎡
⎢
⎣∂y 1
∂t
∂y 2
∂t⎤
⎥
⎦=⎡
⎢
⎣α 11 α 12α 21 α 22⎤
⎥
⎦⎡
⎢
⎣y 1 −y ̄ 1y 2 −y ̄ 2⎤
⎥
⎦+⎡
⎢
⎣R 1R 2⎤
⎥
⎦.The backward-difference approximation to the solution of the system of differential
equations gives the system in EqCM form (see Bårdsen, Hurn and Lindsay, 2004,
for details), namely:
[
%y 1
%y 2]t=[
−α 11 c 1
−α 22 c 2]
+[
α 11 0
0 α 22][
y 1 −δ 1 y 2
y 2 −δ 2 y 1]t− 1+⎡
⎢⎣R 1R 2⎤
⎥⎦t− 1+
1
2[
α 11 α 12
α 21 α 22][
y 1
y 2]t− 1+⎡
⎢
⎣R 1R 2⎤
⎥
⎦
t− 1+
5
12[
α 11 α 12
α 21 α 22][
^2 y 1
^2 y 2]t− 1+⎡
⎢⎢
⎣^2 R 1^2 R 2⎤
⎥⎥
⎦
t− 1+
3
8[
α 11 α 12
α 21 α 22][
^3 y 1
^3 y 2]t− 1+⎡
⎢⎢
⎣^3 R 1^3 R 2⎤
⎥⎥
⎦
t− 1+···,