1. Consider equation (18.15) with
k
5 2. Using the IV approach to estimating the gh
and r, what would you use as instruments for
yt21?
2 An interesting economic model that leads to an econometric model with a lagged dependent variable relates
yt
to the
expected value
of
xt, say,
xpt
, where the expectation is based on all observed information
at time
t
2 1:
yt
5 a0 1 a1xpt
1
ut. [18.68] A natural assumption on 5ut6 is that E1ut
0It21 2 5 0, where
It21 denotes all information on
y
and
x
observed at time
t
2 1; this means that E1yt
0It21 2 5 a0 1 a1xpt
. To complete this model, we need an assumption about how the expectation
xpt
is formed. We saw a simple example of adaptive expectations in Section 11-2, where
xpt
5
xt21. A more complicated adaptive expectations scheme is
Xpt
2
xpt
21 5 l1xt21 2
xpt
21 2, [18.69]
where 0 , l , 1. This equation implies that the change in expectations reacts to whether last period’s realized value was above or below its expectation. The assumption 0 , l , 1 implies that the change in expectations is a fraction of last period’s error.
(i) Show that the two equations imply that
yt
5 la0 1 11 2 l2yt21 1 la1xt21 1
ut
2 11 2 l2ut21. [Hint: Lag equation (18.68) one period, multiply it by 11 2 l2, and subtract this from (18.68).
Then, use (18.69).]
(ii) Under E1ut
0It21 2 5 0, 5ut6 is serially uncorrelated. What does this imply about the new errors,
vt
5
ut
2 11 2 l2ut21?
(iii) If we write the equation from part (i) as
yt
5 b0 1 b1yt21 1 b2xt21 1
vt, how would you consistently estimate the bj?
(iv) Given consistent estimators of the bj, how would you consistently estimate l and a1?