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(a) Consider the ARMA(1,1) process Zt=ϕZt−1+at−θat−1; where ϕ and θ are model parameters, and a1,a2,… are independ
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(a) Consider the ARMA(1,1) process Zt=ϕZt−1+at−θat−1; where ϕ and θ are model parameters, and a1,a2,… are independ
(a) Consider the ARMA(1,1) process Zt=ϕZt−1+at−θat−1; where ϕ and θ are model parameters, and a1,a2,… are independent and identically distributed random variables with mean 0 and variance σa2 (i) Show that the variance of the process is γ0=1−ϕ21−2θϕ+θ2σa2. (5 marks) (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ρk={11−2θϕ+θ2(1−θϕ)(ϕ−θ)ϕk−1 if k=0 if k≥1 (8 marks) (b) Assume that 100 observations from the ARMA(1,1) model gave the following estimates: Find the method of moments estimates of ϕ,θ and σa2. (2+6+3 marks)
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