Consider the following MA(1) model where f"tg es WN(0; 1): yt = "t + _ "t􀀀1: (a) Calculate the autocovariance function of the process. (b) Find the moment estimator of _. (c) Show that the bias of...

Consider the following MA(1) model where f"tg es WN(0; 1):

yt = "t + _ "t􀀀1:

(a) Calculate the autocovariance function of the process.

(b) Find the moment estimator of _.

(c) Show that the bias of this estimator is 􀀀_=n.

(d) Assume that y1 = "1 and de_ne " = ("1"n), y = (y1yn).

Show that we can write Ly = " where L is a lower triangular matrix

and _nd it.

(e) Find the inverse L􀀀1 and verify that the variance-covariance matrix

of y can be written as _ = L􀀀1(L􀀀1)0.

(f) Show that _􀀀1 = L0L.