CS计算机代考程序代写 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,... 22 and comment on the result. 1

Leave a Reply

Your email address will not be published. Required fields are marked *