1. A
partial adjustment model
is
ypt
5 g0 1 g1xt
1
et
yt
2
yt21 5 l1ypt2
yt21 2 1
at,
where
ypt
is the desired or optimal level of
y
and
yt
is the actual (observed) level. For example,
ypt
is
the desired growth in firm inventories, and
xt
is growth in firm sales. The parameter g1 measures the
effect of
xt
on
ypt
. The second equation describes how the actual
y
adjusts depending on the relationship
between the desired
y
in time
t
and the actual
y
in time
t –
1. The parameter l measures the speed of
adjustment and satisfies 0 , l , 1.
(i) Plug the first equation for
ypt
into the second equation and show that we can write
yt
5 b0 1 b1yt21 1 b2xt
1
ut. In particular, find the bj
in terms of the gj
and l and find
ut
in terms of
et
and
at. Therefore, the partial adjustment model leads to a model with a lagged dependent variable and a contemporaneous
x.
(ii) If E1et
0xt,
yt21,
xt21, p2 5 E1at
0xt,
yt21,
xt21, p2 5 0 and all series are weakly dependent, how
would you estimate the bj?
(iii) If b^ 1 5 .7 and b^ 2 5 .2, what are the estimates of g1 and l?