[SOLVED] CS MP, MS, DT.

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 = 0.6Yt1 + Zt 0.3Zt1 0.1Zt2 .
Question 2 By considering the existence of moments show that the process Yt =Zt +a(Zt1 +Zt2 +)
where a is a constant, is non-stationary. Show, however, that the process {Vt} obtained by taking first differences, i.e. Vt = DYt = Yt Yt1, is an MA(1) process and hence stationary. Calculate the autocorrelation function of {Vt}.
Question 3 Given the following MA processes: a) Xt = 0.9t1 + 0.2t2 + t,
b) Xt = 0.3t1 0.6t2 + t,
c) Xt = 1.5t1 + 0.75t2 0.125t3 + t,
where {t} 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 = 0.3Xt1 0.4Xt2 + 1.3t1 + 0.7t2 + t, b) Xt = 1.1Xt1 0.3Xt2 + 1.2t1 + t,
c) Xt = 0.7Xt1 + 0.6Xt2 0.5t1 + 0.4t2 + t, d) Xt = 0.8Xt1 + 0.3Xt2 + 0.6t1 0.5t2 + t,
where t are i.i.d. N(0,1) random variables.
Question 5 Given an AR(1) model: Xt = 1Xt1 + t with |1| > 1, where t are iid with E(t) = 0 and E(2t) = 2. 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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