MATH38032 Time Series Analysis
Examples sheet 3
1. a) What is an AR model? What is an AR process?
b) Is any time series satisfying an AR model an AR process?
c) Under what condition does an AR model have a stationary solution?
d) What is the condition for an AR model to be causal?
2. Find the stationary solution to each model below, where {ε t} is a white noise.
(a) xt + 0.5xt−1 = ε t ,
(b) xt − 0.3xt−1 = ε t + 0.4ε t−1 ,
Also find the best linear predictor of xt in terms of xt−1 , xt−2 , . . . in each case.
3. Suppose the time series {xt} has unit variance and autocorrelations
r(1) = 0.56, r(2) = 0.168, r(3) = 0.0504.
Find the partial autocorrelations at lags 1, 2, and 3 and mean square errors for the best linear predictors using 1, 2, and 3 past values. [Hint: use the Levin-Durbin algorithm. Answers: φ 11 = 0.56, φ22 = −0.2121,
φ33 = 0.0841, σ1(2) = 0.6864, σ2(2) = 0.6555, σ3(2) = 0.6509.]
4. Suppose {ε t} is a white noise and {xt} is the stationary solution to xt − axt−1 = ε t, where | a | > 1. (a) Let et = xt − a −1xt−1 for all t. Show that
et − aet−1 = ε t − a −1ε t−1 .
This means {et} satisfies an ARMA(1,1) model.
(b) Show that the autocovariance function R (s) of {xt} satisfies
R (s) − aR(s − 1) − aR(s + 1) + a2 R (s) = σε(2)δs ,
where σε(2) is the variance of ε t and δs is the delta function.
(c) Find the autocovariance function of {et} in terms of that of {xt} and use it to show that {et} is a white noise with variance σε(2)/a2 . This proves Proposition 1 (week 4).
(d) Write down a causal model for {xt}. Only this step requires | a | > 1.
5. Suppose {ε t} is a white noise and {xt} is the stationary solution to xt − 5xt−1 + 6xt−2 = ε t.
(a) Show that (1− 2B)(1− 3B)xt = ε t.
(b) Let et = (1 − B)(1 − 3/1B)xt. Show that et = (1− 2B)−1 (1 − B)(1− 3B)−1 (1 − 3/1B)ε t.
(c) Use the result in q4(c) or otherwise to show that {et} is a white noise and find its variance.
(d) Write down a causal model for {xt}.