A partial adjustment model is yp t 5 g0 1 g1xt 1 et yt 2 yt21 5 l1yp t 2 yt21 2 1 at

, where yp t is the desired or optimal level of y and yt is the actual (observed) level. For example, yp t is the desired growth in firm inventories, and xt is growth in firm sales. The parameter g1 measures the effect of xt on yp t . 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 l measures the speed of adjustment and satisfies 0 , l , 1.

(i) Plug the first equation for yp t into the second equation and show that we can write yt 5 b0 1 b1yt21 1 b2xt 1 ut

.

In particular, find the bj in terms of the gj and l 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 E1et 0xt

, yt21, xt21, p2 5 E1at 0xt

, yt21, xt21, p2 5 0 and all series are weakly dependent, how would you estimate the bj

?

(iii) If b^

1 5 .7 and b^

2 5 .2, what are the estimates of g1 and l?

8 Suppose that the equation yt 5 a 1 dt 1 b1xt1 1 p 1 bkxtk 1 ut satisfies the sequential exogeneity assumption in equation (11.40).

(i) Suppose you difference the equation to obtain Dyt 5 d 1 b1Dxt1 1 p 1 bkDxtk 1 Dut

.

How come applying OLS on the differenced equation does not generally result in consistent estimators of the bj

?