MATH38032 Time Series Analysis
Examples sheet 8
1. Suppose x1,..., xn is a finite realisation of an ARMA(p,q) process with known p , q.
(a) What is the purpose of decorrelating x1,...,xn?
(b) How do you decorrelate x1,..., xn given the AR and MA parameters?
(c) Is the variance of xt needed in the above?
(d) What is the reduced log-likelihood?
(e) Apart from the innovations algorithm, what other algorithm can be used to calculate e1,...,en?
2. Let {xt} be an ARMA(2,1) process satisfying
(1− 0.3B)(1− 0.4B)xt = (1+ 0.5B)εt.
(a) Find the first three terms after ε t in the expression
xt = εt+ b1 εt−1+ b2 εt−2+ b3 εt−3 + · · ·
by comparing coefficients on both sides of
(1− 0.3z)(1− 0.4z)(1+ b1z + b2z2 + b3z3 + ···) = (1+ 0.5z).
(b) Check your answers using
ARMAtoMA(ar= . . . , ma= . . . , lag.max=3)
3. Find the variance of xt in q2 assuming σε(2) = 1.
4. Fit ARIMA(0,1,1)x(0,1,1)12 , ARIMA(0,1,2)x(0,1,1)12 and any other model you identify from the sample acf and pacf to the air miles data and make a comparison.