11.7 A partial adjustment model is yt *
0  1xt  et yt  yt1
(yt *  yt1)  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 xt is growth in firm sales. The parameter 1 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  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  1yt1  2xt  ut . In 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 ,yt1,xt1,…)
E(at
xt ,yt1,xt1,…)
0 and all series are weakly dependent, how would you estimate the j ? (iii) If ˆ 1
.7 and ˆ 2
.2, what are the estimates of 1 and ?