Consider a model with four lags:

yt = 0 + 0zt + 1zt1 + 2zt2 + 3zt3 + 4zt4 + ut. (1)

Now, assume that the j follow a linear function in the lag, j:

j = 0 + 1j,

for parameters 0 and 1. This is an example of polynomial distributed lag (PDL) model.

Assume that ut = ut1 + et where {et} is an iid process with E(et) = 0 and Var(et) = 1 and a parameter and u0 = 0.

(iv) What are the consequences of this dynamics of ut for the OLS inference in model (1) if = 0 (rho not equal to 0)?

(v) Give an example of test that you would rely on to detect this problem in your model? (You are not expected to give too much details here. The test's inputs and descriptive name are enough.)

(vi) Describe a correction procedure that yields asymptotic valid F and t-tests.

(vii) Assume that ||

(viii) Assume that = 1. Find Var(ut). Is there any chance that {ut} is covariance stationary? Why?