Homework 2:
Exercise 0.0.1. Find the autocovariance of the following ARMA(p,q) processes in the cases where there exists an stationary solution:
1. Φ(B)Xt = Zt
, where Φ(z) = and n ∈ N.
2. Xt = Φ(B)Zt
, where Φ(z) = and n ∈ N.
Exercise 0.0.2. Let Φ(z) and Θ(z) be polynomials of degree 2. Assume that there exists z1, z2 ∈ R such that Φ(z1) = Φ(z2) = 0. Let Φ
′
(z) be the derivative of Φ(z). Assume that Φ
′
(z) and Θ(z) do not share zeroes. Can the ARMA equations Φ
′
(B)Xt = Θ(B)Zt admit a causal solution in the following settings?
1. If z1, z2 ∈ [−1, 1];
2. If z1, z2 ∈ [2, 3]
Exercise 0.0.3. Fin the autocovariance generating function of the following ARMA processes:
1. Xt − 0.5 Xt−1 = Zt;
2. Xt = Zt + Zt−1
Exercise 0.0.4. Denote as B2(R) the set of processes {Xt}t∈Z such that
• Show that the function
is a norm.
• Show that B2(R), ∥ · ∥B2(R)) is a Banach space (Cauchy sequences converge).
• Show that the operator BXt = Xt−1 is a bijection (it is well defined and admits an inverse) on (B2(R), ∥ · ∥B2(R)).
• Let Φ(z) be a polynomial having all its roots in R. Assume that Φ(z) ≠ 0 for all |z| ≤ 1. Show that Φ(B) defines a bijection on (B2(R), ∥ · ∥B2(R)). (Decompose Φ as a product of degree one polynomials and use induction.)
• Under the setting of the previous point, let Φ(B)−1 be the inverse of Φ(B). Show that if Zt
is stationary then Φ(B)−1 Zt
is stationary. Conclude that the equation Φ(B)Xt = Zt admits a unique stationary solution. Is the solution causal?