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TSA701S - TIME SERIES - 1ST OPP - JUNE 2023 |
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nAm I BI A UnlVERS ITY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEALTH, NATURAL RESOURCES AND APPLIED SCIENCES
SCHOOL OF NATURAL AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS, STATISTICS AND ACTUARIAL SCIENCE
I QUALIFICATION:
QUALIFICATION
I CODE:
I COURSE CODE:
I SESSION:
I DURATION:
BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND
STATISTICS
07BAMS
I LEVEL: 7
TSA701S
JUNE 2023
3 HOURS
COURSE
I NAME:
I PAPER:
I MARKS
TIME SERIES ANALYSIS
THEORY
100
1ST OPPORTUNITY
I EXAMINER
I MODERATOR
EXAMINATION QUESTION PAPER
I Dr. Jacob Ong'ala
I Prof. Lilian Pazvakawambwa
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
PERMISSIBLE MATERIALS
Non-programmable calculator without cover
THIS QUESTION PAPER CONSISTS OF 3 PAGERS (including the front page)
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QUESTION ONE - 20 MARKS
The data in the table below shows the exchange rate between the .Japanese yen and the US
dollar from 1984-Ql through 1994-Q4.Use the data shown in the table below to answer the
questions that follow.
Period Actual
Period Actual
Mar-88
Jun-88
Sep-88
Dec-88
Mar-89
Jun-89
Sep-89
Dec-89
Mar-90
Jun-90
Sep-90
Dec-90
124.5
132.2
134.3
125.9
132.55
143.95
139.35
143.4
157.65
152.85
137.95
135.4
Mar-91
Jun-91
Sep-91
Dec-91
Mar-92
Jun-92
Sep-92
Dec-92
Mar-93
Jun-93
Sep-93
Dec-93
140.55
138.15
132.95
125.25
133.05
125.55
119.25
124.65
115.35
106.51
105.1
111.89
(a) Plot the data
(2 mks]
(b) Estimate a triple exponential smoothing model with a smoothing parameter a = 0.6.,
f3 = 0.8. and 'Y = 0.1.
(14 mks]
(c) Plot the smoothing model on the same graph in (a) above
[1 mks]
(d) Compute the mean square error for the model in (b) above
QUESTION TWO - 20 MARKS
[3 mks]
A first order moving average M A(2) is defined by Xt = Zt + 0 1zt-l + 02zt-2 \\Vhere Zt ~
W N(0, a 2 ) and the Zt : t = 1, 2, 3... , T are uncorrelated.
(a) Find
(i) :tvieanof the M A(2)
(ii) Variance of the M A(2)
(iii) Autocovariance of the M A(2)
(iv) Autocorrelation of the M A(2)
[2 mks]
(6 mks]
[8 mks]
[2 mks]
(b) is the MA(2) stationary? Explain your answer
[2 mks]
QUESTION THREE - 22 MARKS
Consider AR(3) :Yi= ¢1Yi.-1 + ¢2Yi.-2 + <f>3Yt.-+2 Et where Et is identically independently
distributed (iid) as white noise.The Estimates the parameters can be found using Yule Walker
equations as
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Ln (Xt - µ)(Xt-h - µ)
Ph=Th = _t=_l _ n _____
_
I:(Xt - µ)2
t=l
'Yo= Var=~ f,(Xt
n t=l
n
µ = L Xt
t=l
- µ)2
Use the data below to evaluate the values of the estimates (¢ 1,<h,¢3 and a-;)
t 1 2 3 4 5 6 7 8 9 10
Xt
26
34 35 38 39
37 38
QUESTION FOUR - 18 MARKS
[22 mks]
Consider the ARMA(l,2) process Xt satisfying the equations Xt - 0.6Xt-l = Zt - 0.4Zt-l -
0.2zt_ 2 Where Zt ~ vVN(0, a-2 ) and the Zt : t = 1, 2, 3..., Tare uncorrelated.
(a) Determine if Xt is stationary
[4 mks]
(b) Determine if Xt is casual
[2 mks]
(c) Determine if Xt is invertible
[2 mks]
(d) Vhite the coefficients W1 of the MA( oo) representation of Xt
QUESTION FIVE - 20 MARKS
[10 mks]
(a) State the order of the following ARIMA(p,d,q) processes
[12 mks]
(i) Yi= 0.8Yt-1 +et+ 0.7et-l + 0.6et-2
(ii) Yi = Yi-1+ et - 0et-l
(iii) Yi= (1 + ef>)Yi--1ef>Yi+-2et
(iv) Yi = 5 + et - ½et-l - ¼et-2
(b) Verify that (max Pl = 0.5 nd min p1 = 0.5 for -oo < 0 < oo) for an MA(l) process:
Xt = E:t- 0c:t-l such that E:tare independent noise processes.
[8 mks]
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