2: Due date: Oct 20 Friday Let at N ID(0, 2 ) if not specified otherwise. 1. Let Xt be a stationary time series with mean and autocovariances k = = P10 Xt /5 0.8k and X t=6 (a) Find E(X). (b) Find Var(X). 2. Consider the process Zt = at + at1 + 0.25at2 , a2 = 20. (a) (b) (c) (d) (e) Identify the order of the ARIMA model for the process. Is {Zt } stationary? Is {Zt } invertible? Find the ACVF (k) and ACF (k) of {Zt } for k = 0, 1, 2, 3,.... Find the values of k , k = 0, 1, 2, 3,... if the process is written as P at = i=0 t Zti . 3. Consider the AR(2) process Zt = 0.5Zt1 0.06Zt2 + at , where at s 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 (0), (1) and (2). 4. Find ACVF (k), k=0,1,2,3,.... of the process Zt = 0.7Zt4 + at . 5. Find the AR and MA representation of the process Zt = 0.6Zt1 + at + 0.2at1 , at W N (0, 4) . 6. Identify the following as specific ARIMA models: a) Zt = 1.5Zt1 0.5Zt2 + at 0.3at1 + 0.6at2 . b) Zt = 3Zt1 3Zt2 + Zt3 + at + 0.1at1 . 7. Consider the ARMA(2,1) model Zt = 0.6Zt1 0.09Zt2 + at 0.2at1 , at W N (0, 1) . a) Find the AR representation of {Zt }. b) Find the ACF (k) of {Zt } for k Z. 1