Gunnar Bårdsen and Ragnar Nymoen 861with:
c 1 =
(
y ̄ 1 +δ 1 ̄y 2)
, δ 1 =
α 12
α 11
c 2 =(
y ̄ 2 +δ 2 ̄y 1)
, δ 2 =
α 21
α 22.As before, the variablesy 1 andy 2 can be considered as stationary functions of
non-stationary components – cointegration is imposed upon the system. Consider
the previous example, assuming linearity soRi=0, and ignoring higher-order
dynamics for ease of exposition:
[
%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
[
%(rw−βz)
^2 z]t=[
−α 11 c 1
−α 22 c 2]
+[
α 11 0
0 α 22][
(rw−βz)−δ 1 z
z−δ 2 (rw−βz)]t− 1,or multiplied out:
%rwt=−α 11 c 1 +α 11 (rw−βz)t− 1 +βzt−α 12 zt− 1zt=−α 22(
y ̄ 2 +
α 21
α 22
y ̄ 1)
+(
α 22 − 1)
zt− 1 −α 21 (rw−βz)t− 1.Ifα 21 =0 and
∣
∣α 22 − 1
∣
∣<1 the system simplifies to the familiar exposition of abivariate cointegrated system withzbeing weakly exogenous forβ:
%rwt=−α 11 c 1 +α 11 (rw−βz)t− 1 +βzt−α 12 zt− 1
zt=−α 22 z ̄+(
α 22 − 1)
zt− 1 ,where the common trend is a productivity trend.
17.2.5 From a discretized and linearized cointegrated VAR representation to
a dynamic SEM in three steps
We will keep this section brief, as comprehensive treatments can be found in many
places – e.g., Hendry (1995a), Johansen (1995, 2006), Juselius (2007), Garrattet al.
(2006), and Lütkepohl (2006) – and only make some comments on issues in each
step in the modelling process we believe merit further attention.
17.2.5.1 First step: the statistical system
Our starting point for identifying and building a macroeconometric model is to
find a linearized and discretized approximation as a data-coherent statistical system
representation in the form of a cointegrated VAR:
yt=c+yt− 1 +∑ki= 1t−iyt−i+ut, (17.14)