STAT 4005 Time Series Assignment 2 Due date: 23 Feb 2023 Let at ~ WN(0, 02) if not specified otherwise. 1. Let Xt be a stationary time series with mean a and autocovariances y = 0.8" and X = Et-6 Xt/5 (a) Find E(X). (b) Find Var(X). 2. Consider the process Zt = at + at-1 + 0.25at-2, Ca = 20. (a) Identify the order of the ARIMA model for the process. (b) Is { Zt } stationary? (c) Is { Zt } invertible? (d) Find the ACVF y(k) and ACF p(k) of { Zt} for k = 0, 1, 2, 3,.... (e) Find the values of TK, k = 0, 1, 2, 3,... if the process is written as at = Li=o TtZt-i . 3. Consider the AR(2) process Zt = 0.5Zt-1 - 0.06Zt-2 + at , where ats are independently and identically distributed as N(0, 1). (a) Find the roots of the AR characteristic equation. (b) Is the process Zt stationary and causal? Why? (c) Find the autocovariances y(0), y(1) and (2). 4. Find ACVF v(k), k=0, 1,2,3,.... of the process Z+ = 0.7%t-4 + at . 5. Find the AR and MA representation of the process Zt = 0.6Zt-1 + at + 0.2at-1, at ~ WN(0, 4) . 6. Identify the following as specific ARIMA models: a) Zt = 1.5Zt-1 - 0.5Zt-2 + at - 0.3at-1 + 0.6at-2. b) Zt = 3Zt-1 - 3Zt-2 + Zt-3 + at + 0.1at-1.7. Consider the ARMA(2,1) model Zt = 0.6Zt-1 - 0.09Zt-2 + at - 0.2at-1, at ~ WN(0, 1) . a) Find the AR representation of { Zt}. b) Find the ACF p(k) of {Zt} for ke Z. 8. Show that for |o| > 1, Zt = at is a white noise process with Var( Z+ ) = $7 . 2