# second-order autoregressive process

Question 1
Consider the second-order autoregressive process
yt = 0 + 2yt−2 + t
where |2| < 1, and t  WN(0, 2). Derive the following: (a) Et−1yt [1 mark] (b) Etyt+2 [1 mark] (c) cov(yt, yt−1) [3 marks] (d) cov(yt, yt−2) [2 marks] Question 2 Consider the following AR(3) process: yt = 0.4yt−1 + 0.5yt−2 − 0.3yt−3 + t Derive the polynomial equation needed to solve for the eigenvalues of the companion matrix for the AR(3) process above. DO NOT SOLVE THE EQUATION. [3 marks] Question 3 Consider the following model: yt = yt−1 + 2t + t (1) Describe how you can make yt stationary. Also show the mathematical proof behind your reasoning. [3 marks] Question 4 Let {yt} be an AR(3) process given by: yt = 1yt−1 + 2yt−2 + 3yt−3 + t (2) (a) Prove that dyt = yt−1 + 1dyt−1 + 2dyt−2 + t (3) where = 1 + 2 + 3 − 1 (4) 1 = −(2 + 3) (5) 2 = −3 (6) In your workings, make sure you clearly state the additional terms that you are adding and subtracting. [3 marks] (b) Derive the autoregressive polynomial associated with the time series defined in (3) and prove that if a unit root exists, then the equation in (4) is equal to 0. [1 mark] Question 5 Consider the following VAR(p) model for N variables: xt = A0 + A1xt−1 + A2xt−2 + ... + Apxt−p + t where t  n.i.d(0, ). If all the matrices of vector autoregressive coefficients (A1,A2, ...,Ap) are diagonal and is diagonal, show and explain what the VAR(p) model will essentially consist of. [3 marks]

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