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FinancialEconometricsHomework1
Dr.Cai
WenboZhang
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1.DownloadweeklypricedataforIBMandMicrosoftstock.IBM(P1t)01/02/6201/15/08
MSFT(P2t)03/13/8601/15/08
a) CreateatimeseriesofcontinuouslycompoundedweeklyreturnsforIBMandMicrosoft.
Usetheequation: ,wecangetthereturnsseries.
b) Use the constructedweekly returns toconstruct a series ofmonthlyreturns. Youmayassumeforsimplicitythatonemonthconsistsoffourweeks.
Use the equation: mi,k = ri,tt=4( k1)+1t=4( k1)+4
, where i =1,2, k=1,2,3, , we can get the
returnsseries.
c) Constructagraphofstockpriceseries(P1t,P2t)andreturnsseries(r1t,r2t).
Time
p1t
0 500 1000 1500 2000
1
00
200
300
400
500
600
Time
p2t
0 200 400 600 800 1000
50
100
150
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Time
p1rv
0 500 1000 1500 2000
0
5000
10000
15000
20000
Time
p2rv
0 200 400 600 800 1000
0
500
1000
1500
Time
r1rm
0 500 1000 1500 2000
0.00
0.05
0.10
Time
r2rm
0 200 400 600 800 1000
-0.06
-0
.04
-0.02
0.00
0.02
0.04
0.06
0.08
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e) What is the definition of a stationary stochastic process? Do prices look likestationaryprocess?Why?Doreturnslooklikeastationaryprocess?Why?
Definition of a stationary stochastic process: In the mathematical sciences, a
stationaryprocessisastochasticprocesswhosejointprobabilitydistributiondoes
notchangewhenshiftedintimeorspace.Asaresult,parameterssuchasthemean
andvariance,iftheyexist,alsodonotchangeovertimeorposition.Asanexample,
whitenoiseisstationary.
Fromtheobservationofthegraphsabove,pricesforbothstocksarenotstationaryprocesssincetherollingmeanischangingovertimeandrollingvarianceincludes
severalhighvalues.
However, the rollingmeanofreturn processes doesnt appear any obvious trend
overtime.Butthebignumbersinvariancedeterminethatreturnprocessesarenot
stationaryeither.
f) Computeautocorrelationcoefficientskfor1k5forpricesandreturnsseries.
P1t[,1]
[1,] 1.0000000
[2,] 0.9952586
[3,] 0.9905095
[4,] 0.9856203
[5,] 0.9807094
[6,] 0.9760011
Time
r1rv
0 500 1000 1500 2000
0.00
0.05
0.10
0.15
Time
r2rv
0 200 400 600 800 1000
0.00
0.01
0.02
0.03
0.04
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P2t
[,1]
[1,] 1.0000000
[2,] 0.9808425
[3,] 0.9619482
[4,] 0.9441063[5,] 0.9262671
[6,] 0.9107144
r1t
[,1]
[1,] 1.000000000
[2,] -0.003720422
[3,] 0.020920001
[4,] 0.015134049
[5,] -0.035417986
[6,] -0.063093877
r2t
[,1]
[1,] 1.000000000
[2,] -0.002964954
[3,] -0.012364290
[4,] 0.015955982
[5,] -0.024635205
[6,] -0.060960500
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Lag
ACF
Series p1t
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Lag
ACF
Series p2t
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g) BasedonthecomputedautocorrelationsforIBMandMSFTstockpricesandreturns,whatcanyousayaboutcorrelationbetweenstockpricesfordifferentdays?What
canyousayaboutcorrelationbetweenstockreturnsfordifferentdays?
Fromthegraphin(f),wecanseethatthestockpricesseriesarehighlycorrelated,
since the correlationcoefficients are very close to1. So, the correlationbetween
stockpricesfordifferentdaysisverystrong.But,thestockreturnsseriesareweakly
correlated,sincethecorrelationcoefficientsareverycloseto0.So,thecorrelation
betweenstockreturnsfordifferentdaysisveryweak.
h) UsingyourstockreturnsforIBMandMSFT,rit,i=1,2,constructfourmoreseriesyit=|rit|,i=1,2and=1,2.Computeautocorrelationcoefficientskfor1k5forthe
newlyconstructedseries.Comparethecomputedcorrelationsfor| rit|,=1,2,and
|rit|.Areresultsasyouexpected?
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Lag
ACF
Series r1t
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Lag
ACF
Series r2t
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Theresultsareasexpected.
i) UsetheJarque-Beratest(seeJarqueandBera(1980,1987))totesttheassumptionofreturnnormalityforIBMandMSFTstockreturns.
JarqueBeraTest
data:r1t
X-squared=6966470,df=2,p-value
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2.UseRprogramtoestimatetheprobabilitydensityfunctionofstandardizedIBMand
MSFTstockreturns.
(a)EstimateandconstructagraphoftheestimatedprobabilitydensityfunctionforIBM
andMSFTstockreturns.
-3 -2 -1 0 1 2 3
0.0
0.2
0.4
0.6
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(b) Onthe same graphwith the empiricaldensity, construct a graphof the standard
normaldensityfunction.Commentyourresults.
Comparedtothestandardnormaldistribution,whichisthegreencurveinthis
graph,thetwoempiricaldistributionshaveahigherpeakandheavy-tail,whichis
normalaccordingtothetextbook.
-3 -2 -1 0 1 2 3
0.0
0.2
0.4
0.6
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(c)ConstructQQ-plotforstandardizedIBMandMSFTreturns.YoumayusetheR
commandforthis.Commentyourresults.
AccordingtothetwoQQplotgraphsabove,itsonlysimilartostandardnormal
distributionfrom-2to1.5forIBMstockreturn.Andits-1.5to1.5forMSFTstock
return.
-3 -2 -1 0 1 2 3
0
5
10
15
20
25
Normal Q-Q Plot
Theoretical Quantiles
SampleQuantiles
-3 -2 -1 0 1 2 3
-2
0
2
4
6
8
10
Normal Q-Q Plot
Theoretical Quantiles
SampleQuantiles
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Rcodes
Problem1
library(fTrading)
library(tseries)
IBM=read.csv(file="/Users/brianzwb/Documents/Financial
Econometrics/IBM.csv",header=T,skip=1)
MSFT=read.csv(file="/Users/brianzwb/Documents/Financial
Econometrics/MSFT.csv",header=T,skip=1)
IBMprice=IBM[,5]
IBMreturn=diff(log(IBMprice))
MSFTprice=MSFT[,5]
MSFTreturn=diff(log(MSFTprice))
MSFTreturn
r1t=IBMreturn
r2t=MSFTreturn
dim(r1t)=c(4,length(r1t)/4)
monthly1=colSums(r1t)
r1t=IBMreturn
dim(r2t)=c(4,length(r2t)/4)
r2t=r2t[1:1136]
dim(r2t)=c(4,length(r2t)/4)
monthly2=colSums(r2t)
r2t=MSFTreturn
p1t=IBMprice
p2t=MSFTpricets.plot(p1t)
ts.plot(p2t)
ts.plot(r1t)
ts.plot(r2t)
p1rm=rollMean(p1t,13)
p1rv=rollVar(p1t,13)
p2rm=rollMean(p2t,13)
p2rv=rollVar(p2t,13)
r1rm=rollMean(r1t,13)
r1rv=rollVar(r1t,13)r2rm=rollMean(r2t,13)
r2rv=rollVar(r2t,13)
ts.plot(p1rm)
ts.plot(p2rm)
ts.plot(p1rv)
ts.plot(p2rv)
ts.plot(t1rm)
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ts.plot(r1rm)
ts.plot(r2rm)
ts.plot(r1rv)
ts.plot(r2rv)
acf(p1t,5)$acfacf(p2t,5)$acf
acf(r1t,5)$acf
acf(r2t,5)$acf
absr1t=abs(r1t)
absr2t=abs(r2t)
absqr1t=r1t^2
absqr2t=r2t^2
acf(absr1t)$acf
acf(absr1t,5)$acf
acf(absr2t,5)$acfacf(absqr1t,5)$acf
acf(absqr2t,5)$acf
jarque.bera.test(r1t)
jarque.bera.test(r2t)
Problem2
d1={r1t-mean(r1t)}/sd(r1t)
d2={r2t-mean(r2t)}/sd(r2t)
x0=seq(-3,3,length=100)
y0=density(d1,n=100,from=-3,to=3)
y1=y0$y
qqnorm(d1)
qqline(d1,col=2)
y2=density(d2,n=100,from=-3,to=3)
y3=y2$y
qqnorm(d2)
qqline(d2,col=2)
matplot(x0,cbind(y1,y3),type="l",lty=1:3,xlab="",ylab="")
matplot(x0,cbind(y1,y3,dnorm(x0)),type="l",lty=c(1,2),xlab="",ylab="")