MAY 2019 EXAMINATION DIET
SCHOOL OF MATHEMATICS & STATISTICS
MODULE CODE: MT4527
MODULE TITLE: Time Series Analysis
EXAM DURATION: 2 hours
EXAM INSTRUCTIONS: Attempt ALL questions.
The number in square brackets shows the
maximum marks obtainable for that
question or part-question.
Your answers should contain the full
working required to justify your solutions.
A formulae sheet is provided at the end of
the exam.
PERMITTED MATERIALS: Non-programmable calculator
YOU MUST HAND IN THIS EXAM PAPER AT THE END OF THE EXAM
PLEASE DO NOT TURN OVER THIS EXAM PAPER UNTIL YOU ARE
INSTRUCTED TO DO SO.
MT4527 May 2019, Page 1 of 12
1. (a) Explain in simple words what the state-space of a stochastic process Xt
is. [1]
(b) Consider a random walk Xt = Xt1 + “t, t = 0, 1, 2, . . ., with initial value
X0 = 0. Here, “t is a white noise stochastic process.
(i) Derive the expectation, E(Xt), for all t = 1, 2, 3, . . . [1]
(ii) Derive the variance, V ar(Xt), for all t = 1, 2, 3, . . . [2]
(iii) Derive the covariance, Cov(Xt, Xt+h), for all h = 0, 1, 2, . . .. Hence,
derive the correlation Corr(Xt, Xt+h), for all h = 0, 1, 2, . . .. [3]
(c) Consider a Simple Moving Average (SMA) estimator for the trend compo-
nent of a non-seasonal model with trend. The choice of span q is ad hoc,
and subject to the trade-o between two of the estimator’s properties.
Name those two
properties. [2]
2. Consider a stationary AR(p) process with general form,
Xt = 1Xt1 + 2Xt2 + · · ·+ pXtp + “t, t = 0,±1,±2, . . .
where “t denotes a white noise stochastic process. Derive the Yule-Walker
equations for the calculation of the autocorrelation sequence for the AR(p)
process, including the initial conditions. [3]
MT4527 May 2019, Page 2 of 12
3. Let “t denote a white noise process. Consider the AR(2) process {Xt} given
by,
Xt +
1
3
Xt1 2
9
Xt2 = “t.
(a) Show that this stochastic process is stationary by considering and solving
the relevant equation (x) = 0, where (x) represents the AR operator in
a standard ARMA model. [1]
(b) Substitute the general expression for the MA(1) process into the defining
equation of the AR process, and show that the coecients of the MA(1)
process satisfy a recurrence relation, specifying the recurrence relation and
the initial conditions. [4]
(c) Hence, calculate the coecients of the infinite MA representation. [4]
(d) Assume without proof that, for L > 0, the minimum mean square error
forecast x t(L), based at origin t, of Xt+L satisfies the equation
Xt+L = 0″t+L + 1″t+L1 + . . .+ L1″t+1 + x t(L).
Use this equation to show that x t(L) equals the conditional mean of Xt+L
given the information available at time t. Show also that the corresponding
conditional variance of Xt+L equals 2″