[SOLVED] CS 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 + 1Zt1 + 2Zt2
(ii) the MA(3) process Yt = Zt + 1Zt1 + 2Zt2 + 3Zt3 Plot the autocorrelation for:
(i) MA(2): 1 = 0.8, 2 = 0.5
(ii) MA(3): 1 = 0.8, 2 = 0.4, 3 = 0.3
Question 2 For the MA(1) process Yt = Zt + Zt1, find the maximum and minimum values of 1 and the values of for which they are attained. Do the same for 1 and 2 (with reference to 1 and 2) for the MA(2) process.
Question 3 Check that all of the following AR(2) processes are causal stationary: a)S Xt = 1.4Xt1 0.65Xt2 + t,
b)S Xt = 0.45Xt1 + 0.25Xt2 + t,
c) Xt = 1.2Xt1 0.75Xt2 + t,
where t i.i.d. with E(t) = 0 and Var(t) = 2. Calculate and display (k), k = 0, 1, 2, , 9.
Question 4 Consider the AR(2) process Yt = 1Yt1 + 2Yt2 + Zt. Determine 1 and 2 in terms of 1 and 2 and viceversa.
Question 5 (harder question)
1. Show that the AR(2) process Yt = 0.5Yt1 + 0.14Yt2 + Zt is stationary, and that the
acf {k} is given by:
k = 17 (0.2)k + 112(0.7)k, k = 0,1,2,. 129 129
Plot {k} for k 0.
2. Show that the AR(2) process Yt = 0.6Yt2 + Zt is stationary, and that the acf {k} is
given by:
k =1ik(0.6)k/2{1+(1)k} k=0,1,2, 2
Plot {k} 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 YuleWalker equations is of the form = azk + bzk. k12
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