An interesting economic model that leads to an econometric model with a lagged dependent variable relates yt to the expected value of xt , say, xp t , where the expectation is based on all observed information at time t 2 1:
A natural assumption on 5ut 6 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 a1xp t . To complete this model, we need an assumption about how the expectation xp t is formed. We saw a simple example of adaptive expectations in Section 11-2, where xp t 5 xt21. A more complicated adaptive expectations scheme is
EQUATION 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
EQUATION 18.68
[Hint: Lag equation (18.68) one period, multiply it by 11 2 l2, and subtract this from (18.68). Then, use (18.69).]
Under E1ut 0It21 2 5 0, 5ut 6 is serially uncorrelated. What does this imply about the new errors, vt 5 ut 2 11 2 l2ut21?