An interesting economic model that leads to an econometric model with a lagged dependent variable relates yt to the expected value of xt, say, x*i, where the expectation is based on all observed information at time t - 1:

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) = α0 + αl x*1. To complete this model, we need an assumption about how the expectation x*t is formed. We saw a simple example of adaptive expectations in Section 11.2, where x*t = xt-1. A more complicated adaptive expectations scheme is

where 0 (i) Show that the two equations imply that

(ii) Under E(ut | It-1) = 0, {ut} is serially uncorrelated. What does this imply about the new errors, vt = ut - (1 - λ) ut-1?
(iii) If we write the equation from part (i) as how would you consistently estimate the βj?
(iv) Given consistent estimators of the βj, how would you consistently estimate A and α1?

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