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Times Series Past Paper

this document provides some practices on times series

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28 views5 pages

Times Series Past Paper

this document provides some practices on times series

Uploaded by

samsonths9487
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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M3/M4/M5 S8 Imperial College London BSc, MSci and MSc EXAMINATIONS (MATHEMATICS) May - June 2011 This paper is also taken for the relevant examination for the Associateship of the Royal College of Science. Time Series Date: Wednesday, 01 June 2011. Time: 200pm. Time allowed: 2 hours. This paper has FOUR questions. Candidates should use ONE main answer book ‘Supplementary books may only be used after the main book is full Statistical tables will not be provided. * DO NOT OPEN THIS PAPER UNTIL THE INVIGILATOR TELLS YOU TO. «Answer all the questions. Each question carries equal weight # Credit will be given for all questions attempted, but extra credit will be given for complete or nearly complete answers. Affix one of the labels provided to each answer book that you use, but DO NOT USE THE LABEL WITH YOUR NAME ON IT. # Calculators may not be used © 2011 Imperial College London 3/Ma/M5 S8 Page 1 of 5 Note: Throughout this paper {«,} is a sequence of uncorrelated random variables (white noise) having zero mean and variance o2, unless stated otherwise. The unqualified term “stationary” will always be taken to mean second-order stationary. All processes are real-valued unless stated otherwise. The sample interval is unity unless stated otherwise 1. (2) (i) What is meant by saying that a stochastic process is stationary? (ii) Let {¥4} be a stationary process with mean zero, and define X= (Bo + Bite + Ye where Jp, 8; are constants and 1 is a deterministic seasonal component with period 12. Compute W, = (1 — BY?)?X,, where B is the backward shift operator, and determine whether {1V;} is a stationary process or not. (b) A continuous-time process {X(t)}, with t in seconds, has spectral density function he Wis, 0, otherwise, Sxeolf) with C a constant and f in cycles/second. It is sampled with sample-interval At = (1/2) second to produce the discrete-time process {X,}. What is the spectral density function of {X,}? (©) Consider the stationary moving average process {X,} defined by ¢ 3 2 gat pena t gtt-2 Xi ore 3 (i) Determine whether or not {X;} is invertible. (ii) Find the transfer function (or frequency response function) G(f) corresponding to the linear time-invariant digital filter L{-} defined by 2 5 2 Heda tert beat id (iii) Hence show that the spectral density function of {X+} is given by gee taki su) = [B+ geostann)] 2 M3S8/M4S8 Time Series (2011) Page 2 of 5 2. (a) (i) Suppose {X;} is an MA(g) process with zero mean, ie., Xz can be expressed in the form Xe = ~Ooiges — Brgeena— 6 Seati-ar where the @,,'s are constants (6, sequence is given by = -1,€q #0). Show that its autocovariance a= [EXIT aPssrt@ rsa, 0, if jr] > @. (ii) Let {2%} bea stationary, zero mean, normal/Gaussian process with autocorrelation sequence px.-. Show that the autocorrelation sequence for ¥; = X? is given by Prix = Pe [Hint: if A and B are two jointly normal/Gaussian random variables with zero mean then B{A?B*) = E{A?}B{B?} + 2B7{AB}] (iii) Find a model for a stationary, zero mean, normal/Gaussian process {X,}, such that ¥; = X} has autocovariance sequence {sy} given by 200, r=0; 0, 18, 9, giving all parameter combinations that satisfy the stated form of {sy, (b) Let {X;} be defined by Xt = bi2Xe-1 + G2aXi-2 + ty t= 2,3,4,... where Xo =X; =0. Find var{X;} for t = 2,3 and 4 in terms of 12 and $o. M3S8/M4SB Time Series (2011) Page 3 of 5 Let X1,....Xw be a sample of size N from a white noise (uncorrelated) process with (a) unknown mean j1 and variance o. The so-called ‘unbiased’ autocovariance estimator is given by Noh : DY & ~ A) (Kea - X. i (i) Show that, for 0 < |r|

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