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.
