Find the ad of this process - when m 1 use sas to simulate

Question

Problems:

1. Consider the MA(m)-process, with equal weights 1/(m+1) at all lags, given by

X_t = Σ^m_k=01/(m+1) Z_t k

(a) Find the ad of this process.

(b) When m = 1 use SAS to simulate a realization of this process.

2. Consider the following models where {Zt} is a Gaussian white noise.

X_t = -.98 X_t 1 + Z_t

X_t = -.98_Z-1 + Z_t

X_t = .6 X_t-i - 1.2Z_t-1 + Z_t

X_t = X_t-1 + .8 X_t-2 - .5 Z_t-1 + Z_t

(a) Find the acf and plot it for k = 0,1, ....,10

(b) Simulate a series of 100 observations from each of the models with σ_z² = 1 and plot the sample acf for k = 0,....20.

3. Express the models in Problem 2 using the B operator.

4. Suppose that (X_t) is a stationary process with autocovariance function γx. Express the autocovariance function of the difference filter of first order ∇X_t = X_t - X_t-1 in terms of γx. Find it when γx(k) = λ^|k|.

5. Prove or disprove the following process is weakly stationary:

(a) Z_t = A sin(2Πt + θ) where A is a constant, and θ is a random variable that is uniformly distributed on [0,2Π].

(b) Z_t = A sin(2Πt + 0) where A is a random variable with zero mean and unit variance, and θ is a constant.