**Question 1.**

Consider the time series {Xt}t∈Z

Xt = β0 + β1t + β2t

2 + Wt, where {Wt}t∈Z ∼ WN(0, σ2), and β0, β1, β2 are real constants with β1 ̸= 0 and β2 ̸= 0.

(a) Show that both Xt and ∇Xt are non-stationary.

(b) Show that ∇2Xt is stationary.

**Question 2.**

Assume that {Xt} satisfies the equation

(1 − ϕB)Xt = Wt,

(a) Generate n = 300 observations from the process (1) with ϕ = 0.35. Plot the simulated

time series and its ACF. Provide your R code. Does this generated time series look stationary?

Justify your answer

(b) Now repeat (a) with ϕ = 0.999. Plot the simulated time series and its ACF. Does this generated time series look stationary? Justify your answer

**Question 3.**

Let {Xt}t∈Z be a time series satisfying the representation Xt = − X∞ j=1 ϕ −jWt+j , t ∈ Z, (2) for |ϕ| > 1, where {Wt}t∈Z ∼ WN(0, σ2 ).

(a) Write Equation (2) in terms of backshift operators and compute the mean and variance functions of Xt Hint: B−jYt = Yt−(−j) = Yt+j

(b) Compute the autocovariance of {Xt}t∈Z and argue that this process is stationary.

**Question 4. **

Let {Yt}t∈Z be a stationary process with mean function µ and autocovariance

function cov(Yt+h, Yt) = γY (h), for h ∈ Z. Define

Xt = ∇dYt = Yt − Yt−d, t ∈ Z,

where d ∈ N. Compute the mean function and the autocovariance function of {Xt}t∈Z in terms of γY (·). Is {Xt}t∈Z stationary? Justify your answer.

**Question 5.**

Let {Wt}t∈Z ∼ WN(0, σ2). Identify the AR and MA polynomials and determine

whether the following ARMA processes are causal and/or invertible. Also, watch out for parameter redundancy.

(a) Xt = −0.2Xt−1 + 0.48Xt−2 + Wt

(b) Xt = −1.9Xt−1 − 0.88Xt−2 + Wt + 0.2Wt−1 + 0.7Wt−2.