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MATH38032 Time Series Analysis
Examples sheet 6
1. a) In a seasonal ARMA model for monthly time series, what is the purpose of having a seasonal AR part and a seasonal MA part? Think about dependence on past values.
b) What is the advantage of having seasonal AR and MA parts in the model compared with using an ordinary ARMA model? Think about the number of parameters.
c) What does an ARIMA(p,d, q) model for {xt} say about {(1− B)dx t}?
d) What does an ARIMA(p,d, q)x(P, D, Q)s model for {xt} say about {(1− B)d (1− Bs )Dx t}?
2. a) Expand (1− B)3 and apply it to xt.
b) Difference t3 once, twice and three times.
c) Multiply out (1− φB)(1− B)2 and (1+θB)(1+ ΘB4).
d) Expand the ARIMA(1,2,1)x(0,0,1)4 model in ARMA(p, q) form.
3. Find the autocorrelation function of a stationary time series {xt} satisfying
(a) xt = (1− 0.5B)(1− 0.4B4)ε t ,
(b) (1− 0.5B)xt = (1− 0.4B4)ε t ,
where {εt} is a white noise. Want answers for all lag s ∈ Z.
4. Use ARMAacf to compute the acf at lags up to lag 20 for the time series in q3.
acf <- ARMAacf(ar= . . . , ma= . . . , lag.max=20)
plot(acf, type="h")
abline(h=0)
5. (a) Use ‘arima.sim’ to simulate n = 500 observations from the AR(1) model
(1− 0.5B)xt = εt ,
where {ε t} is a Gaussian white noise with mean 0 and variance 1. (b) Plot the data and the sample acf up to lag 40.
(c) Difference the data if nonstationary and repeat (b).
(d) Comment on the results in (b) and (c).
x <- arima .sim(model=list(order=c(p,d,q), ar= . . . , ma= . . .), n=500) par(mfrow=c(2,1))
plot(x, type="l") #that is the letter ell
acf(x, lag.max=40)
x <- x[2:500] - x[1:499] #or x <- diff(x)
etc
Repeat the above with the AR(2) model
(1−1.5B +0.5B 2 )xt = εt ,
where {εt} is a Gaussian white noise with mean 0 and variance 1.