A partial adjustment model is
y*i = (0 + (1xt + et
yt - y t-1 = ((y*t - yt-1) + at,
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 x, is growth in firm sales. The parameter y, measures the effect of x, on y*. 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 ( measures the speed of adjustment and satisfies 0 < ( < 1.
(i) Plug the first equation for yt* into the second equation and show that we can write
yt = (0 + (1yt-1 + (2xt + ut.
ln particular, find the (j in terms of the (j and ( 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.
(ii) If E(et|xt,yt-1, xt-1,...) = E(at|xt,yt-1,_xt-1,...) = 0 and all series are weakly dependent, how would you estimate the Bj?
(iii) If 1 = .7 and 2 = .2, what are the estimates of (1 and (?