CS计算机代考程序代写 MP, MS, DT.

MP, MS, DT.
F70TS2 – Time Series
Exercise Sheet 2 – Moving Average and Autoregressive Processes
Question 1 Determine the autocorrelation function for: (i) the MA(2) process Yt = Zt + β1Zt−1 + β2Zt−2
(ii) the MA(3) process Yt = Zt + β1Zt−1 + β2Zt−2 + β3Zt−3 Plot the autocorrelation for:
(i) MA(2): β1 = 0.8, β2 = 0.5
(ii) MA(3): β1 = 0.8, β2 = −0.4, β3 = −0.3
Question 2 For the MA(1) process Yt = Zt + βZt−1, find the maximum and minimum values of ρ1 and the values of β for which they are attained. Do the same for ρ1 and ρ2 (with reference to β1 and β2) for the MA(2) process.
Question 3 Check that all of the following AR(2) processes are causal stationary: a)S Xt = −1.4Xt−1 − 0.65Xt−2 + εt,
b)S Xt = 0.45Xt−1 + 0.25Xt−2 + εt,
c) Xt = 1.2Xt−1 − 0.75Xt−2 + εt,
where εt i.i.d. with E(εt) = 0 and Var(εt) = σε2. Calculate and display ρ(k), k = 0, 1, 2, …, 9.
Question 4 Consider the AR(2) process Yt = α1Yt−1 + α2Yt−2 + Zt. Determine ρ1 and ρ2 in terms of α1 and α2 and vice–versa.
Question 5 (harder question)
1. Show that the AR(2) process Yt = −0.5Yt−1 + 0.14Yt−2 + Zt is stationary, and that the
acf {ρk} is given by:
ρk = 17 (0.2)k + 112(−0.7)k, k = 0,1,2,…. 129 129
Plot {ρk} for k ≥ 0.
2. Show that the AR(2) process Yt = −0.6Yt−2 + Zt is stationary, and that the acf {ρk} is
given by:
ρk =1ik(0.6)k/2{1+(−1)k} k=0,1,2,… 2
Plot {ρk} for k ≥ 0.
Hint: for an AR(2) processes with characteristic polynomial with roots z1 and z2 outside the
unit circle, the solution of the Yule–Walker equations is of the form ρ = az−k + bz−k. k12
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