CS计算机代考程序代写 MP, MS, DT.
MP, MS, DT.
F70TS2 – Time Series
Exercise Sheet 1 – Stationarity and the autocorrelation function
Question 1 Let {εt} be iid random variables with E(εt) = 0 and var (εt) = σε2 (White Noise). We define a process X by
Xt = βεt−1 + εt with |ψ| < 1.
Show that X is weakly stationary and that the autocorrelation function of X has the form
1 , ρX(k) = ρ(±1),
k = 0 ,
k = ±1, (1) |k| > 1.
0,
Calculate ρX(±1) in terms of β and show that −1 < ρX(±1) < 1.
Question 2 Let {εt} be a white noise. Calculate the acf of the following processes: a) Xt = 0.5εt−1 + 0.4εt−2 + εt,
b) Xt = 0.8εt−1 − 0.2εt−2 + εt,
c) Xt = 0.6εt−1 + 0.3εt−2 − 0.2εt−3 + εt,
Question 3 By considering first and second order moments (i.e. means, variances, covari- ances), investigate whether or not each of the following processes is (weakly) stationary. As always, {Zt} is a white noise process.
(i) Yt =Yt−1 +Zt
(ii) Yt =Yt−1 +α+Zt (α̸=0)
(iii) Yt =αYt−1 +Zt (|α|<1)
(iv) Yt = Zt−1Yt−2 + Zt, where σZ2 = 1
Question 4 Consider a time series process {Yt} that is the sum of two independent stationary processes {Ut} and {Vt}, so Yt = Ut + Vt. Show that {Yt} is a stationary process.
Question 5 (Hard question) Consider a stationary process {Ut} with acf {ρUk }. Ut represents a “signal”, but because of noise/measurement error/interference on Ut, what we actually observe is {Yt}, where Yt = Ut + Zt. It may be assumed that the signal and the noise are independent processes. Let the “signal to noise ratio” be defined as SNR = σU2 /σZ2 .
Show that {Yt} is a stationary process with acf {ρYk } given by:
1−1 ρYk = 1+SNR
ρUk, k=1,2,3,...
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and comment on the result.
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