1. Consider an MA(6) model with ₁ = 0.5, ₂ = 0.25, ₃ = 0.125, ₄ = 0.0625, ₅ = 0.03125, and ₆ =

0.015625. Find a much simpler model that has nearly the same -weights.

2 Let {Yt} be an AR(2) process of the special form Yt = ₂Y_{t 2} + e_t. Use first principles

to find the range of values of ₂ for which the process is stationary.

3 Consider the AR(1) model Y_t = Y_{t 1} + e_t. Show that if || = 1 the process cannot

be stationary. (Hint: Take variances of both sides.)

4 Consider an MA(7) model with 1 = 1, 2 = 0.5, 3 = 0.25, 4 = 0.125,

5 = 0.0625, 6 = 0.03125, and 7 = 0.015625. Find a much simpler model that

has nearly the same -weights.

5 Consider the model Y_t = e_{t 1} e_{t 2} + 0.5e_{t 3}.

(a) Find the autocovariance function for this process.

(b) Show that this is a certain ARMA(p,q) process in disguise. That is, identify

values for p and q and for the 's and 's such that the ARMA(p,q) process

has the same statistical properties as {Yt}.