MP, MS, DT.
F70TS2 – Time Series
Exercise Sheet 3 – MA(∞), AR(∞), ARMA and ARIMA
Question 1 Calculate the autocorrelation function of the ARMA(1,2) process Yt = 0.6Yt−1 + Zt − 0.3Zt−1 − 0.1Zt−2 .
Question 2 By considering the existence of moments show that the process Yt =Zt +a(Zt−1 +Zt−2 +…)
where a is a constant, is non-stationary. Show, however, that the process {Vt} obtained by taking first differences, i.e. Vt = DYt = Yt − Yt−1, is an MA(1) process and hence stationary. Calculate the autocorrelation function of {Vt}.
Question 3 Given the following MA processes: a) Xt = −0.9εt−1 + 0.2εt−2 + εt,
b) Xt = 0.3εt−1 − 0.6εt−2 + εt,
c) Xt = −1.5εt−1 + 0.75εt−2 − 0.125εt−3 + εt,
where {εt} is a WN. Show that all of these processes are invertible.
Question 4 Find out which of the following ARMA processes are causal stationary and/or invertible, which are neither causal stationary nor invertible.
a) Xt = 0.3Xt−1 − 0.4Xt−2 + 1.3εt−1 + 0.7εt−2 + εt, b) Xt = 1.1Xt−1 − 0.3Xt−2 + 1.2εt−1 + εt,
c) Xt = 0.7Xt−1 + 0.6Xt−2 − 0.5εt−1 + 0.4εt−2 + εt, d) Xt = 0.8Xt−1 + 0.3Xt−2 + 0.6εt−1 − 0.5εt−2 + εt,
where εt are i.i.d. N(0,1) random variables.
Question 5 Given an AR(1) model: Xt = φ1Xt−1 + εt with |φ1| > 1, where εt are iid with E(εt) = 0 and E(ε2t) = σε2. Show that there is a stationary MA(∞) representation for this process with absolutely summable coefficients. What is the special feature of this process?
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