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Statistics - Time Series Level H
创建日期：2024-05-04 14:41:41
所属分类：统计学statistics
标签： Statistics
Time Series Level H

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Statistics - Time Series Level H
This paper consists of 5 pages and contains 3 question(s).
Candidates should attempt all questions.
Question 1 20 marks
Question 2 20 marks
Question 3 20 marks
Total 60 marks
The following material is made available to you:
Statistical tables∗
Statistical Tables
Formula sheet
“An electronic calculator may be used provided that it is allowed under the School of
Mathematics and Statistics Calculator Policy. A copy of this policy has been distributed
to the class prior to the exam and is also available via the invigilator.”
NOTE: Candidates should attempt all 3 questions.
1
CONTINUED OVERLEAF/
1. (a) Given a real data set, how could you determine if the correlation structure would
be well modelled by an MA(q) process? [2 MARKS]
(b) Autoregressive processes of orders 1, 2, and 3 were fitted to a time series with 120
observations, and the model output from R is shown below.
arima(x, order = c(1, 0, 0))
Coefficients:
ar1 intercept
0.9363 287.9945
s.e. 0.0160 38.0214
arima(x, order = c(2, 0, 0))
Coefficients:
ar1 ar2 intercept
0.7597 0.1901 289.9881
s.e. 0.0439 0.0441 46.6512
arima(x, order = c(3, 0, 0))
Coefficients:
ar1 ar2 ar3 intercept
0.7415 0.1184 0.0953 290.6858
s.e. 0.0445 0.0553 0.0447 51.6608
What type of AR(p) process is appropriate here? That is, what is the value of p?
Explain your answer. [3 MARKS]
(c) Consider the following AR(1) and MA(1) processes A and B.
A : Xt = 0.3Xt−1 + Zt,
B : Xt = 0.6Zt−1 + Zt.
i. Describe how correlation is induced between Xt and Xt−1 for processes A and
B. [2 MARKS]
ii. For each process, what is the value of ρ2, the lag two autocorrelation coeffi-
cient?
[2 MARKS]
(d) Consider the time series process
Xt = 0.7Zt−1 − 0.25Zt−2 + Zt,
2
CONTINUED OVERLEAF/
where Zt is a purely random process with E [Zt] = 0 and Var [Zt] = σ2z .
i. What type of process is defined here? [1 MARK]
ii. Is the process stationary, invertible, or both? Explain your answer.
[4 MARKS]
(e) Consider the model
Xt = β0 + β1t+ β2t
2 + λ1Zt−1 + λ2Zt−2 + Zt,
where β0, β1, β2, λ1 and λ2 are all non-zero and finite, and Zt is a purely random
process with E [Zt] = 0 and Var [Zt] = σ2z .
i. Calculate the mean and variance of Xt. Is it weakly stationary? Explain your
answer. [4 MARKS]
ii. Let Yt = ∇Xt be the first order difference of Xt. Write out Yt as a function
of Zt. [2 MARKS]
2. (a) Consider the time series process Xt given by
Xt = β0 + β1t+ Yt,
where Yt is a weakly stationary AR(1) process with lag one coefficient α = 0.7.
i. What would be a naive way to estimate β1, assuming that the error process
Yt is independent? How could this be done in R? [1 MARK]
ii. Give one drawback of the naive approach from i. in the estimation of β1
compared with a more robust approach that allows for the correlation present
in Yt. How could a more robust approach be performed in R? [2 MARKS]
iii. As part of a simulation study you may want to assess parameter estimation.
Detail two standard metrics that can be used to measure the quality of pa-
rameter estimates.
[2 MARKS]
(b) Consider the time series process given by
Xt = exp
(
β0 + β1t+ β2t
2 + St + Zt
)
,
where Zt is a purely random process with E [Zt] = 0 and Var [Zt] = σ2z , and St is
a seasonal component with period d, such that St = St+d. Show how this process
can be made weakly stationary. [5 MARKS]
3
CONTINUED OVERLEAF/
(c) The autocorrelation function (ACF) and partial autocorrelation function (PACF)
have been computed for a time series with 700 observations and are shown in the
table below.
Lag τ 1 2 3 4 5 6 7 8 9 10
ACFτ 0.57 0.42 0.28 0.10 0.08 0.03 -0.01 0.01 0.01 -0.01
PACFτ 0.76 0.68 0.12 0.06 0.05 0.03 0.020 0.01 0.01 0.01
Would you prefer an AR(p) or an MA(q) model for this process? If so, what would
be the corresponding order of the chosen model? [3 MARKS]
(d) Given below is R code to simulate time series data.
time <- 1:500
beta0 <- 0.3
beta1 <- 0.55
Z.sd <- 10
Y <- arima.sim(list(order=c(1,0,0), ar=0.6), n=500, sd=Z.sd)
X <- beta0 + beta1 * time + Y
Write out the time series model in clear mathematical notation. Provide details
on any trend, seasonality, or short-term correlation induced by the model.
[7 MARKS]
3. (a) Consider the ARIMA(1, 1, 0) process. Is this a stationary process? [2 MARKS]
(b) Two ARIMA processes are given below:
A : ARIMA(3, 2, 1)
B : ARIMA(1, 0, 2)
Write out both processes A and B in backshift notation. Hence write out the
model equations for both processes, expressing Xt in terms of Zt and previously
occurring quantities. [4 MARKS]
(c) Consider the time series model
Xt = β0 + β1t+ β2t
2 + Zt t = 1, . . . , 30,
where Zt is a purely random process with E [Zt] = 0 and Var [Zt] = σ2z . This
model was fitted to data with 30 time points which provided estimated regression
parameters βˆ0 = 4, βˆ1 = 0.8, βˆ2 = 0.6, and an estimated error variance of σˆ
2
z = 1.2.
i. Calculate the one-step ahead prediction x30(1) for time t = 31. [2 MARKS]
4
CONTINUED OVERLEAF/
ii. Calculate the 95% prediction interval for x30(1). [6 MARKS]
(d) Consider the AR(2) model
Xt = 0.75Xt−1 − 0.125Xt−2 + Zt,
where Zt is a purely random process with E [Zt] = 0 and Var [Zt] = σ2z = 2, and
xn = −1, xn−1 = 3, and xn−2 = 2. Calculate the one, two and three steps ahead
forecasts and their associated error variances. [6 MARKS]
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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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