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TSA701S - TIME SERIES - 2ND OPP - JULY 2023 |
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nAmlBIA UnlVERSITY
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
STATISTICS
07BAMS
I LEVEL: 7
MATHEMATICS
AND
TSA701S
JULY 2023
3 HOURS
COURSE
I NAME:
I PAPER:
I MARKS
TIME SERIES ANALYSIS
THEORY
100
SUPPLEMENTARY/
I EXAMINER
I MODERATOR
2ND OPPORTUNITY 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
(a) Verify that (max p1 = 0.5 nd min p1 = 0.5 for -oo < 0 < oo) for an MA(l) process:
Xt = ct - 0ct-1 such that ct are independent noise processes.
[8 mks]
(b) State the order of the following ARIMA(p,d,q) processes
(i) Yi= (1 + </>)Yi--1 </>Yi-2+ et
(ii) Yi = Yt-1 + et - 0e1.-1
(iii) Yi = 5 + et - ½et-1 - ¼et-2
(iv) Yt = 0.8Yi-1 +et+ 0.7et-l + 0.6et-2
QUESTION TWO - 22 MARKS
[12 mks]
Consider AR(3) :Yi= </>1Yt-1+ </>2Yt.-+2 <f>3Yt+.-2ct where ct is identically independently
distributed (iid) as white noise.The Estimates the parameters can be found using Yule Walker
equations as
where
Ln (X,. - µ)(Xt-h - µ)
Ph= rh = _t=__l_ n_____
_
I;(Xt - µ)2
t=l
'Yo= Var=~ I:(Xt
n t=l
n
µ = L Xt
t=l
- µ)2
Use the data below to evaluate the values of the estimates (</>1(,P2<,f>a3nd O';)
t 1 2 3 4 5 6 7 8 9 10
Xt 13 17 15 14 19 22 20 26 32 32
QUESTION THREE - 20 MARKS
[22 mks]
A first order moving average M A(2) is defined by Xt
WN(0,0' 2 ) and the Zt: t = 1,2,3 ...,T are uncorrelated.
(a) Find
(i) Mean of 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)
+ + Zt 01zt-1 02Zt-2 Where Zt ~
[2 mks]
[6 mks]
[8 mks]
[2 mks]
(b) is the MA(2) stationary? Explain your answer
[2 mks]
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QUESTION FOUR - 18 MARKS
Consider the ARMA(l,1) process Xt satisfying the equations Xt - 0.5Xt-l = Zt + 0.4zt-l
Where Zt ~ vVN(O, cr2 ) and the Zt : t = 1, 2, 3... , T are uncorrelated.
(a) Determine if Xt is stationary
(b) Determine if Xt is casual
(c) Determine if Xt is invertible
(d) Write the coefficients \\J.11of the MA(oo) representation of X 1.
QUESTION FIVE - 20 MARKS
[4 mks]
[2 mks]
[2 mks]
[10 mks]
Use the following data shown in the table below to answer the questions that follow.
It 1112131
41 516111
81 91101
I Yi I 1 I 8 I 9 I 11 I 13 I 15 I 14 I 13 I 18 I 11 I
Given Xt =mt+ Rt such that Rt-iS the random component following a white noise with a
mean of zero and variance of cr2 and mt- 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 detrendecl 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]
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