(a) Consider a series {2,} of the form xt = Σvjet-j j=0 i. Consider forecasting In+ based on observations {n-1....}. Giv

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(a) Consider a series {2,} of the form xt = Σvjet-j j=0 i. Consider forecasting In+ based on observations {n-1....}. Give expres- sions for the k-step ahead predictor of Intk and the associated error enk n≥ 1 and k ≥ 1. ii. Give an expression for the variance of the prediction error Vk. [1] iii. Show that Ink+1-ink=(+1-j)en+k-j- juk [2] iv. Consider the MA (2) process given by 1₁ =₁0₁e₁-1-0₂-2, where {e} is white noise: - Find &nk for k 21 in terms of (01.02} and {en, en-en-2,...} [3] - Find the prediction error V₁ for k21 in terms of (0₁,0₂) and o [3] (b) Consider the autoregressive moving average ARMA(1,1) model of {x} of the form I₁-0₁-1=₁-0₁-1 where {e} is a white noise series. i. Express this model in backward operator notation and state the conditions on the parameters and under which this time series model is stationary and invertible. [3] ii. Calculate E(re) and Var(r) when the process is stationary. [6] iii. Assuming invertibility, express this model as an infinite-order AR process * = a + Σπιτικ k=1 and find a general form for the coefficients {} in terms of 8, o and k. iv. Assuming stationarity, express this model as an infinite-order MA process x₁ = ₁ + $xei-k k=1 and find a general form for the coefficients {} in terms of 0, and k. [4]