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TSA701S - TIME SERIES ANALYSIS - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS
| QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND
STATISTICS
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QUALIFICATION 07BAMS
| LEVEL: 7
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CODE:
| COURSE CODE: _ TSA701S
NCAOMUER:SE TIME SERIES ANALYSIS
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| SESSION:
JUNE 2022
| PAPER: THEORY
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| DURATION:
3 HOURS
| MARKS 100
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FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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| EXAMINER
| Dr. Jacob Ong’ala
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| MODERATOR
| Prof. Lilian Pazvakawambwa
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INSTRUCTION
1. Answer all the questions
2. Show clearly all the steps in the calculations
3. All written work must be done in blue and black ink
THIS
QUESTION
PERMISSIBLE MATERIALS
Non-programmable calculator without cover
PAPER CONSISTS OF 3 PAGERS (including the front page)
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QUESTION ONE - 20 MARKS
A first order moving average MA(2) is defined by X; = z + 012%-1 + 02%-2 Where y% ~
WN(0,o7) and the x :t = 1,2,3...,T are uncorrelated.
(a) Find
(i) Mean of the M7 A(2)
(ii) Variance of the MA(2)
(iii) Autocovariance of the MW A(2)
(iv) Autocorrelation of the MA(2)
[2 mks]
[6 mks]
[8 mks]
[2 mks]
(b) is the MA(2) stationary? Explain your answer
[2 mks]
QUESTION TWO - 20 MARKS
Use the following data shown in the table below to answer the questions that follow.
Given X; =m; + R; such that R;-is the random component following a white noise with a
mean of zero and variance of o? and m- is the trend,
(a) Estimate the trend using exponential smoothing method with a smoothing parameter
a = 0.68.
[5 mks]
(b) Estimate the trend using a centred moving average of order 4
[6 mks]
(c) On the same axes, draw the graphs of the original set of data, detrended data (using the
exponential smoothing ) and detrended data (using the moving average)
[7 mks]
(d) Which estimates in (a) or (b) above is a better estimate above ((a) or (b)) and why. [2 mks]
QUESTION THREE - 22 MARKS
Consider AR(3) :¥; = $1Yi:-1 + 62Yi-2 + $3Yi-2 + €; where e& is identically independently
distributed (iid) as white noise.The Estimates the parameters can be found using Yule Walker
equations as
$1
d
Jj=l{
La
mn 1
e\\
pi
[a
p2 } and
o? =$3 yo[(1 — 62p2— 6p3i — 6il3) — 2¢0(d1P3 + $3)91 — 2616392]
where
h
dm — 2)(Xi-n — b)
Ph = Th =
o>
(Xt - pw)?
t1
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Yo* = Var = =NLM t¥=1\\(Xi—p)?
w= dXnm
t=1
Use the data below to evaluate the values of the estimates (¢1, $2, ¢3 and o2)
t] 1} 2] 3] 4] 5] 6] 7] 8] 9] 10
X,}| 13} 17 | 15 | 14] 19 | 22 | 20 | 26 | 32 | 32
QUESTION FOUR - 18 MARKS
[22 mks]
Consider the ARMA(1,1) process X; satisfying the equations X; — 0.5X;-1 = 2 + 0.42-1
Where 3% ~ WN(0, a?) and the z= :¢=1,2,3...,7 are uncorrelated.
(a) Determine if X; is stationary
[4 mks]
(b) Determine if X; is casual
[2 mks]
(c) Determine if X; is invertible
[2 mks]
(d) Write the coefficients UV; of the MA(oco) representation of X;
[10 mks]
QUESTION FIVE - 20 MARKS
(a) Verify that (max p; = 0.5 nd min p; = 0.5 for —oo < @ < oo) for an MA(1) process:
X= €; — Oe4_-1 such that e; are independent noise processes.
[8 mks]
(b) State the order of the following ARIMA(p,d,q) processes
(i) Y= (14+ 6) — OM-2 + et
(ii) Y¥; = Yin + e¢ — Oe¢_-1
(iii) Y,=5+e- $et-1 _ $er—2
(iv) Y¥; = 0.8¥i4 + e¢ + 0.7e¢_1 + 0.6e4_2
[12 mks]