Please help me with the 7 questions..!

Let at NID(0, 2 ) if not specified otherwise. 1. Let Xt be a stationary time series with mean and autocovariances k = 0.8 k and X = P10 t=6 Xt/5 (a) Find E(X). (b) Find Var(X). 2. Consider the process Zt = at + at1 + 0.25at2 , 2 a = 20. (a) Identify the order of the ARIMA model for the process. (b) Is {Zt} stationary? (c) Is {Zt} invertible? (d) Find the ACVF (k) and ACF (k) of {Zt} for k = 0, 1, 2, 3,.... (e) Find the values of k, k = 0, 1, 2, 3,... if the process is written as at = P i=0 tZti . 3. Consider the AR(2) process Zt = 0.5Zt1 0.06Zt2 + 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 (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.

Assignment 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