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MT4527 Time Series Analysis
Creation date：2024-04-10 15:52:48
Belonging category：数学mathematics
Tag MT4527
Time Series Analysis

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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 xˆt(L), based at origin t, of Xt+L satisfies the equation
Xt+L = ⌫0"t+L + ⌫1"t+L1 + . . .+ ⌫L1"t+1 + xˆt(L).
Use this equation to show that xˆt(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"

⌫20 + ⌫
2
1 + . . .+ ⌫
2
L1

. [4]
(e) Suppose now that the random shocks are Normally distributed and that,
after 80 observations have been seen, the point forecast of X82 made at
origin t = 80 is xˆ80(2) = 60. Evaluate the 95% prediction interval for X82
made at the same origin, if the estimate of the white noise variance is
s2" = 81. You may find the information below helpful:
> qnorm(0.975)
[1] 1.959964
[3]
MT4527 May 2019, Page 3 of 12
4. A statistician is asked to model the changes in temperature (measured in the
Farhenheit scale) during a chemical experiment. He has a time series x of 200
measurements taken every hour. For example, a measurement of xt = 10
describes a drop of 10 oF from hour t 1 to hour t.
(a) The statistician plots the original time series (x) as well as the first (rx),
the second (r2x) and the third (r3x) order di↵erenced series - see plot
below. Decide based on the plot the likely minimum order of di↵erencing,
necessary to obtain a stationary time series. Briefly justify your answer. [2]
0 50 100 150 200
−4
0
−2
0
0
Time
x
Original time series
0 50 100 150 200
−4
0
2
4
Time
∇
x
First order differences
0 50 100 150 200
−2
0
2
4
Time
∇
2 x
Second order differences
0 50 100 150 200
−6
−2
2
6
Time
∇
3 x
Third order differences
(b) The statistician decides to fit an ARIMA(0, 1, 2) model using R. The fol-
lowing output is produced:
Call:
arima(x = temp, order = c(0, 1, 2))
Coefficients:
ma1 ma2
0.5074 0.8856
s.e. 0.0351 0.0367
sigma^2 estimated as 1.143: log likelihood = -298.72, aic = 603.43
MT4527 May 2019, Page 4 of 12
The statistician wants to use this model for forecasting. Write down,
without using any backwards operators, an explicit di↵erence equation for
Xt+L, where L = 1, 2, 3, . . . . [2]
(c) Recall that the minimum mean square error forecast xˆt(L) for Xt+L, based
at origin t, equals the conditional mean of Xt+L given the information
available at time t. Hence write down, without proof, a set of di↵erence
equations giving (for L = 1, 2, 3, ..) the forecast xˆt(L) of Xt+L. [3]
(d) Let the forecast origin be t = 200. The observed values for the temperature
changes xt, for t = 198, 199, 200, together with the corresponding estimated
random shocks ✏t, were as follows:
day t xt ✏t
198 -22.24 0.036
199 -24.54 -0.600
200 -25.03 -0.220
Calculate the minimum mean square error forecast of the change in tem-
perature for t = 202. [4]
(e) Consider the ARIMA(0,1,2) model (1 B)Xt = (1 + 0.5B + 0.9B2)"t.
Express the (Xt)t2Z process in random shock form. [4]
MT4527 May 2019, Page 5 of 12
5. A (G)ARCH model was fitted to the first order di↵erenced series (rx) of the
observations on the changes in temperature, as described in Question 4 above.
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 1.723e+00 3.335e+00 0.517 0.605
a1 1.519e-01 1.061e-01 1.432 0.152
a2 4.519e-02 2.806e-01 0.161 0.872
b1 9.293e-15 1.840e+00 0.000 1.000
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 1.8717, df = 2, p-value = 0.3923
Box-Ljung test
data: Squared.Residuals
X-squared = 0.065631, df = 1, p-value = 0.7978
Answer the following questions:
(a) Which type of (G)ARCH model has been fitted? [1]
(b) Write down the volatility equation of the fitted model. [1]
(c) Considering the fitted model, is the variance of the white noise process
finite? If so, compute an estimate for this variance. [2]
(d) Given the provided R output, comment on whether the fitted model is
appropriate for the (rx) data. [3]
MT4527 May 2019, Page 6 of 12
Formulae Sheet
General solutions of first and second order homogeneous recur-
rence relations
• First order homogeneous recurrence relation
uk+1 uk = 0
for k = 0, 1, . . . .
Solution:
uk = A
k, where A is a constant
• Second order homogeneous recurrence relation
uk+2 + c1uk+1 + c2uk = 0
for k = 0, 1, . . . with auxiliary equation:
x2 + c1x+ c2 = 0
Solution
– when the roots of the auxiliary equation, 1 and 2, are distinct:
uk = A1
k
1 + A2
k
2, where A1, A2 are constants
– when the auxiliary equation has a double root :
uk = (A1 + kA2)
k where A1, A2 are constants
MT4527 May 2019, Page 7 of 12
MT4527: Time Series - Solutions
1. (a) The state space of Xt is the set of values that the random variables Xt
may take. [1]
[EASY - bookwork]
(b) (i) For a random walk Xt, without drift and initial value X0 = 0,
Xt = "t + "t1 + ...+ "1
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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