MATH38032 Time Series Analysis
Examples sheet 4
1. a) How do we check the causality of an AR(2) model without finding the roots of a polynomial?
b) What can be said about the autocorrelation function (acf) of an AR(2) process? What about the partial acf?
c) Is any time series satisfying an MA model an MA process?
d) Is εt in an MA model uncorrelated with xt−1 , xt−2 ,. . . ?
e) Does the answer in d) imply that εt is the innovation associated with {xt} at time t?
2. Find the stationary solution to each model below, where {εt} is a white noise with mean 0.
(a) xt − 0.8xt−1 + 0.16xt−2 = εt , t ∈ Z.
(b) xt + 1.5xt−1 − xt−2 = εt , t ∈ Z.
What is the best linear predictor of xt in terms of xt−1 , xt−2 , . . . in each case?
3. Find the acf and pacf of {xt} in q2 assuming stationarity.
4. Find the best linear predictor of xt given xt−1 , xt−2,. . . (infinite past) when
(a) xt = ε t − 0.3ε t−1 − 0.4ε t−2 , t ∈ Z,
(b) xt − 0.3xt−1 = ε t + 0.4ε t−1 , t ∈ Z,
(c) xt − 0.3xt−1 = ε t + 2.5ε t−1 , t ∈ Z,
where {εt} is a white noise and {xt} is stationary.