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ECMT3150: Assignment

Creation date：2024-05-21 15:02:28

Belonging category：计算机computer science

Tag ECMT3150

Assignment

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ECMT3150: Assignment

1. [Total: 24 marks]
Note: Please append your R codes (as a separate .R le) for part (g) while you submit the
assignment.
Let Xi denote the log-price of a stock, Cherry Inc. (code: CRRY), by the end of trading day
i, and let Xi := Xi Xi1; thus Xi is the log-return on trading day i (i.e., over period
(i 1; i]).
Assume fXig_i0 follows the AR(1) model:
Xi = 0 + 1Xi1 + ui: (1)
where ui iid normal with mean 0 and variance 2.
Let fFigi0 be the natural ltration generated by fuigi0.
(a) [2 marks] Express Xi in terms of Xi1 and ui.
(b) [2 marks] Compute E(XijFi1).
(c) [2 marks] Compute V ar(XijFi1).
(d) [2 marks] What is the condition on 0 and 1 such that fXigi1 is a martingale
di¤erence sequence?
A trading strategy is dened by figi0, where i is measurable with respect to Fi. Speci-
cally, i represents the number of CRRY shares a trader buys at the start of day i.
The log-return due to the trading strategy over period (0; T ] is given by
rT =
TX
i=1
i1Xi.
(e) [4 marks] Alice invested in a share of CRRY using a buy-and-hold strategy, with i 1
for all i. Compute E(rT ) and V ar(rT ) with 0 = 0 and 1 = 1.
(f) [4 marks] Bob suggested another strategy, with i Xi for i > 0 and Compute E(rT )
and V ar(rT ) with 0 = 0 and 1 = 1.
1
(g) [8 marks] Carol suggested yet another strategy, with i 1fXi > 0g and 0 = 1.
We want to evaluate the risk-return tradeo¤ of the proposed strategies using computer
simulation.
Start an R session, and set a random seed equal to the last 3 digits of your student ID.1
Then generate B sample values of rT (name them as r
(1)
T ; r
(2)
T ; : : : ; r
(B)
T ), and compute
the sample mean and variance of rT as follows:
rT =
1
B
BX
b=1
r
(b)
T ;
se(rT ) =
1
B 1
BX
b=1
(r
(b)
T rT )2:
For the purpose of your simulations, set T = 63, 2 = 0:1, B = 1000.
The Sharpe ratio, dened as SR = rTse(rT ) , is a common measure of the risk-return
tradeo¤. Trading strategies with higher SR are more preferred by investors.
Complete the following table with SR values. Comment on the performance of the
trading strategies under di¤erent scenarios.
0 1 Alice Bob Carol
0 1
0:01 1
0:01 1
0 0:9
0 1:1
2. [Total: 16 marks] LetM denote the mood of Mimi (h: happy; a: angry), and let W denote
the weather (s: sunny; r: rainy). The joint probability distribution of M and W is given in
the table below. The row and column sums are displayed in the last column and in the last
row, respectively.
p(m;w) M = h M = a
W = s 0:4 0:1
W = r 0:2 0:3
(a) [2 marks] Compute P (M = a).
(b) [2 marks] Derive the conditional distribution of W given M = a.
Assume that, given m and w, your test score S follows a normal distribution with mean
(m;w) := E(SjM = m;W = w) and standard deviation 5. The conditional mean function
(m;w) is given in the table below:
1This is to ensure that your answers are replicable but di¤erent from those of other students.
2
(m;w) m = h m = a
w = s 80 50
w = r 70 40
The passing score is 50 or above.
(c) [3 marks] Compute the mean score E(S).
(d) [3 marks] Given that Mimi was angry, what is the mean score you would get?
(e) [3 marks] Compute the probability of failing the test.
(f) [3 marks] Given that you failed the test, what is the probability that Mimi was angry?
3. [Total: 20 marks]
Note: Please append your R codes (as a separate .R le) while you submit the assignment.
Carol, an amateur economist, proposes the following time series model for unemployment
rate:
yt =
1
20
+
p
3
2
yt1 1
4
yt2 + "t; (2)
where "t iid N(0; 0:022) (normal distribution with mean 0 and variance 0:022). The time
period is measured in number of quarters.
(a) [3 marks] Show that the time series fytg generated by model (1) is stationary.
(b) [3 marks] There is a stochastic cycle in the time series generated by model (1). Find its
periodity in number of quarters.
(c) [4 marks] Compute the ACF for the rst 3 lags, i.e., (1), (2) and (3).
(d) [2 marks] Write an R program to simulate a sample path of fytg over 30 years. Set the
initial values y0 and y1 to be y0 = 0:1 and y1 = 0:12. While simulating the random
numbers for "t, set the random seed to be your last 3 digits of your student ID.
(e) [2 marks] Plot the sample ACF and record its value for the rst 3 lags (the values can be
retrieved from the acf command output stored as a list). Why are they di¤erent from
your answers in part (c)?
(f) [3 marks] Using the simulated sample path in part (d), estimate an AR(2) model using
the R command arima. Write down the estimated model with the parameter estimates
and their standard error. Also record the estimated variance of the innovations.
(g) [3 marks] Using the simulated sample path in part (d) and the R package forecast,
plot the point forecast and the condence interval for each period over the next 5 years.
Describe the short-run and long-run behaviour of the point forecast and the condence
interval.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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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 PHIL2617 PHYS3116 MIE250 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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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