Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 3

An interesting economic model that leads to an econometric model with a lagged dependent variable relates yt to the expected value of xt, say, xt*, where the expectation is based on all observed information at time t - 1:
yt = 0 + a1xx* + ut.
A natural assumption on {ut} is that E(ut|It-1) = 0, where It-1 denotes all information on y and x observed at time t - 1; this means that E(yt|It-1) = 0 + 1xt*. To complete this model, we need an assumption about how the expectation x* is formed. We saw a simple example of adaptive expectations in Section 11.2, where x* = xt-1. A more complicated adaptive expectations scheme is
xt* - xt-1* = (xt-1 - × t-1*)
where 0 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 1 implies that the change in expectations is a fraction of last period's error.
(i) Show that the two equations imply that
yt = 0 + (1 - )yt-1 + 1xt-1 + ut - (1 - )ut-1.
[Hint: Lag equation one period, multiply it by (1 - ), and subtract this from. Then, use.]
(ii) Under E(ut|It-1) = 0, {ut} is serially uncorrelated. What does this imply about the new errors, vt = ut - (1 - )ut-1
(iii) If we write the equation from part (i) as yt= 0+ 1yt-1+ 2xt-1+vt how would you consistently estimate the j
(iv) Given consistent estimators of the j., how would you consistently estimate and 1 Equation yt = 0 + a1xx* + ut.
xt* - xt-1* = (xt-1 - × t-1*)