1. Consider an 𝐴𝑅𝑀𝐴 process given by the equation
𝑋𝑡 = 3+0.4𝑋𝑡−1 + 𝑒𝑡 −0.2𝑒𝑡−1
(i) State the Order of the ARMA Process
(ii) Calculate the mean (assuming it is stationary)
(iii) Determine whether the time series is
(a) Invertible and/or stationary
(b) Purely in deterministic
(c) Markov
(iv) Determine the ACF and PACF
[1]
[6]
[2]
[2]
[6]
2.Consider an autoregressive process (AR) given by the equation;
𝑋𝑡=6−0.7 𝑋𝑡−1+0.2𝑋𝑡−2+𝑒𝑡
(i) State the order of the Process.
[1]
(ii) Calculate the mean (assuming it is stationary)
[2]
(iii) Determine whether the time series is
(a) Invertible and/or stationary
(b) Purely in deterministic
[4]
[2]
(iv)
Markov
[2]
(vi)
Determine the ACF and PACF
[6]
3. (a) 𝑌𝑡, 𝑡 = 0, 1, 2,… is a time series defined by
𝑌 𝑡 −0.8𝑌 𝑡−1 = 𝑍𝑡 +0.2 𝑍𝑡−1
where, {𝑍𝑡} represents a set of uncorrelated random variables with mean 0 and variance 𝜎2. Derive the
autocorrelation𝜌𝑘, 𝑘=0, 1, 2…
[6]
4. Show that the autocorrelation function of the stationary zero mean 𝐴𝑅𝑀𝐴 (1, 1) process:
𝑋𝑡 = 𝛼𝑋𝑡−1 + 𝑒𝑡 + 𝛽𝑒𝑡−1 is given by:
𝜌1 = (1+𝛼𝛽)(𝛼+𝛽) /(1+𝛽2+2𝛼𝛽),
𝜌𝑘 = 𝛼𝑘−1𝜌1 , 𝐾 = 2,3 [6]
5. Consider the second order autoregressive process
𝑋𝑡 = 5−0.4𝑋𝑡−1 +0.05𝑋𝑡−2 +0.2 𝑒𝑡
(a) Determine whether the process can be stationary
[3]
(b) State, with a reason, whether the process possesses the Markov property. [2]
(c) Assuming that 𝜎 = 2, calculate the values of 𝛾0, 𝛾1, 𝛾2.
[10]