A partial adjustment model iswhere yt* is the desired or optimal level of y, and yt is the...

A partial adjustment model is

where yt* is the desired or optimal level of y, and yt is the actual (observed) level. For example, yt* is the desired growth in firm inventories, and xt is growth in firm sales. The parameter γ1 measures the effect of xt on yt*. 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 β measures the speed of adjustment and satisfies 0 ≺ β ≺ 1.

(i) Plug the first equation for yt* into the second equation and show that we can write

In particular, find the βj in terms of the γj and β 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.