ANSWERS TO EXERCISE 9.11
 
 
a:
LQ sub t = beta sub 1 + beta sub 2 LP sub t + beta sub 3 LY sub t +
u sub t.
 
u sub t = rho sub 1 u sub t-1 + rho sub 2 u sub t-2 + rho sub 3 u sub t-3
          + rho sub 4 u sub t-4 + epsilon sub t
 
 H sub 0: rho sub 1 = rho sub 2 = rho sub 3 = rho sub 4 = 0.

b.
Step 1
Regress LQ against 0, LP, and LY, and get u hat sub t = LQ sub t -
beta hat sub 1 - beta hat sub 2 LP sub t - beta hat sub 3 LY sub t.

Step 2.  Generate u hat sub t-1, u hat
sub t-2, u hat sub t-3, and u hat sub t-4.
Step 3
Regress u hat sub t against u hat sub t-1, u hat sub t-2, u hat sub
t-3, u hat sub t-4, constant, LP sub t, and LY sub t, \fIusing only
observations 5 through n\fR.
Step 4
Compute (n-4) R sup 2 where n = no.  of observations and R sup 2 is
unadjusted R sup 2 from (b.3).  Under H sub o, (n-4) R sup 2 is
distributed as chi square with 4 d.f.
Step 5
Reject H sub o if P [ chi sub 4 sup 2 > (n-4) R sup 2 ] < level of
significance or if (n-4) R sup 2 > chi sub 4 sup 2 (*), the point on
chi sub 4 sup 2 such that the area to the right is equal to the level
of significance.

c.
Step 1
Regress LQ against 0, LP sub t, and LY sub t.
Step 2
Get u hat sub t = LQ sub t - beta hat sub 1 - beta hat sub 2 LP sub t -
beta hat sub 3 LY sub t.
Step 3
Generate u hat sub t-1, u hat sub t-2, u hat sub t-3, and u hat sub
t-4.
Step 4
Regress u hat sub t against u hat sub t-1, u hat sub t-2, u hat sub
t-3, and u hat sub t-4 [Note: no constant term here and no LP sub t or
LY sub t ] and get rho hat sub 1, rho hat sub 2, rho hat sub 3, rho hat
sub 4.
Step 5
Generate
 LQ sub t sup * = LQ sub t - rho hat sub 1 LQ sub t-1 -...
                  - rho hat sub 4 LQ sub t-4 
 
 LP sub t sup * = LP sub t - rho hat sub 1 LP sub t-1...
                  - rho hat sub 4 LP sub t-4 
 
 LY sub t sup * = LY sub t - rho hat sub 1 LY sub t-1...
                  - rho hat sub 4 LY sub t-4 

Step 6
Regress LQ sub t sup * against 0, LP sub t sup *, and LY sub t sup *
and get the next round of estimates beta hat sub 1, beta hat sub 2, and
beta hat sub 3.
Step 7
Go back to Step (2) and iterate until ESS for Step (6) doesn't change
by more than some pre-specified number of percentage.
 
d.
Estimates are biased, but consistent, and \fIasymptotically\fR
efficient.