Question

QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of φ2 for which the process is stationary? (ii) Show that if o2-1 the process cannot be stationary

Answer 1

Firstly we multiply the ARMA (L,L) by at and take expectatian Tf Zt is stationary proce ss, Var (Zt)E Var (2t-1 2

For ARMA (1,1) t he coeff. уч 2j k. 1-20002

% consider AR(2) model The causality Condition is Ợ(T)tD kore 2 Causa The process to be easuel-root ф(T),should lie outside the unit circle ie. IZI2L φ Iz2I>L

The condition for sta omary solution to e xists That is, if and mly if φ(z)=1-φ zto for z=ti dL!ะเ The ARMA (Lu) 's not station. ry