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MT4527 Time Series Analysis

创建日期：2024-04-10 15:52:48

所属分类：数学mathematics

标签： 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

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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