Discus the important characteristics of the acf for an arma

Question

EXERCISE 1. 

Are the following series stationary?

1)   Y_t = Y_t-i + ε_t.

2)   y_t = ε_i + ε_t.

3)   y_t = ε_i - θ₁ε_t-i - θ₂e_t-i.

4)    y_t = sin(Πt) + εt.

EXERCISE 2.

Manually sketch (by-hand) the autocorrelation function (ACF) for the following processes :

1) AFtMA(1,1) with Φ = 0.7 and θ = 0.4

2) ARMA(1,1) with Φ = 0.7 and θ = -0.4

Discus the important characteristics of the ACF for an ARMA(1,1) model

EXERCISE 3.

Consider the following AREvIA (p, d, q) time seres :

y_t = 0.8y_t-1- 0.2y_t-2 + εt.

1) Identify p, d and q

2) Calculate the autocovariance function of the saies using the Yule-Walker equations

EXERCISE 4.

Suppose that {xt} is a time series such that : x_t = μ + yt where μ is a constant and the estimator of μ is x' = 1/n Σ_t=1ⁿx_t

Compute :

1) Var(x') if y_t is white noise

2) Var(x') if y_t is ε_t - (1/3)e_t-1

3) Var(x') if y_t is ε_t + (1/3)y_t-1