Texts: QUESTION 3
[Total marks=20]
a) Given that if X is a stationary time series with spectrum fxw and Y = aX with a < 0, then the spectrum of Y is given by f(w) = |a|^2 * fxw. Determine the power spectral density function of an MA(2) process X = 1 - 0.8B + 0.5B^2Z, where B is a backward shift operator and Z ~ N(0,1).
(10)
b) Determine the power spectral density function of an MA(2) process X = 1 - 0.8B + 0.5B^2Z, where B is a backward shift operator and Z ~ N(0,1), as a Fourier Transform of the autocovariance function. (10)
QUESTION 4
[Total marks=10]
Suppose that {X} and {Y} are independent stationary series each with autocovariances Yx and k, respectively.
a) Find the autocovariance of {X+Y}. (5)
b) Find the spectral density of {X+Y} and comment on your answer. (5)