Time Series analysis

Simulation of the action of fundamental trader

and consequent formation of the time series of prices

pf<-1 # fundamental price, intrinsic value (price)

p<-(pf+0.01) # initial price setting

vp<-p # initial vector setting for plot

Ns<-2000 # number of steps

time=c(0:Ns)

for(i in 1:Ns){

if(p>pf) p<-(p * 0.99) # deterministic reduction

if(p<pf) p<-(p * 1.01) # deterministic increase

p<-(p*exp(rnorm(1,mean=0,sd=0.01))) # randomization

vp<-cbind(vp,p)

}

vp<-as.vector(vp)

plot(vp,type="l")

a) acf(vp)

b) pacf(vp)

c) library(tseries)

adf.test(vp)

> adf.test(vp)

Augmented Dickey-Fuller Test

data: vp

Dickey-Fuller = -11.5254, Lag order = 12, p-value = 0.01

alternative hypothesis: stationary

Warning message:

In adf.test(vp) : p-value smaller than printed p-value

(reject null: unit root)

d) kpss.test(vp)

> kpss.test(vp)

KPSS Test for Level Stationarity

data: vp

KPSS Level = 0.0899, Truncation lag parameter = 10, p-value = 0.1

Warning message:

In kpss.test(vp) : p-value greater than printed p-value

(not enough to reject null: i.e. stationary)

Samuelson theory of business cycles - simulation

b3<-1.04

b2<-0.2

b1<-1.01

Y1<-1.0

Y2<-1.04

Y0<-1

Nt<-500

vY<-rep(0,Nt)

for(i in 1:Nt){

Y2<-Y1

Y1<-Y0

Y0<-(b1*(Y1-Y2)+b2*Y1+b3) + 1.0*rnorm(1,mean=0,sd=0.3)

vY[i]<-Y0

}

plot(vY,type="l")

a1) repeat for b1<-0.1 (there is no more "explosive" behaviour)

Linear Filters

plot(UKDriverDeaths)

fi1<-filter(UKDriverDeaths,c(1/2,rep(1,11),1/2)/12)

fi2<-filter(UKDriverDeaths,c(1/2,rep(1,5),1/2)/6)

lines(fi1,col=2)

lines(fi2,col=3)

a1) construct the comprehensive plot including action of

various filters, as wel as generate jpg file as an output

datU<-UKDriverDeaths

f1<-filter(datU,c(1/2,rep(1,5),1/2)/6)

f2<-filter(datU,c(1/2,rep(1,11),1/2)/12)

f3<-filter(datU,c(1/2,rep(1,17),1/2)/18)

f4<-filter(datU,c(1/2,rep(1,23),1/2)/24)

jpeg(filename = "filter4plot.jpg")

par(mfrow=c(2,2))

plot(datU)

lines(f1,col=1)

plot(datU)

lines(f2,col=2)

plot(datU)

lines(f3,col=3)

plot(datU)

lines(f4,col=4)

dev.off()

Decompose

a0)

d1<- decompose(log(UKDriverDeaths))

plot(d1)

a1) plot(co2)

a2)

d2< -decompose(log(co2))

plot(d2)

c-I)

plot(d2$random)

c-II)

d2r<-d2$random

d2s<-d2$seasonal

check: plot(d2r), plot(d2s)

c-III)

calculate acf from d2r and d2s and compare, but there will be some problems

(here are reported problems with NAs when applied adf.test())

d2rr<-d2r[10:450]

d2ss<-d2s[10:450]

> adf.test(d2rr)

Augmented Dickey-Fuller Test

data: d2rr

Dickey-Fuller = -11.7102, Lag order = 7, p-value = 0.01

alternative hypothesis: stationary

Warning message:

In adf.test(d2rr) : p-value smaller than printed p-value

>

> kpss.test(d2rr)

KPSS Test for Level Stationarity

data: d2r

KPSS Level = 0.0067, Truncation lag parameter = 4, p-value = 0.1

Warning message:

In kpss.test(d2rr) : p-value greater than printed p-value

> acf(d2rr)

( still 6 months "memory" exists in random component)

ARIMA MODEL SIMULATION

(find best model, see example from fundamental price generation)

pfun<-1 # "fundamental" price

p<-(pfun+0.01) # initial price setting

vp<-p # auxilliary initial vector setting for plot

Ns<-1000 # number of steps

time=c(0:Ns) # vector of times

for(i in 1:Ns){

if(p>pfun) p<-(p * 0.99) # deterministic reduction

if(p<pfun) p<-(p * 1.01) # deterministic increase

p<-(p*exp(rnorm(1,mean=0,sd=0.01))) #randomization

vp<-cbind(vp,p)

}

# ( to check data by means of plot(time,vp,type="l") )

vp<-as.vector(vp)

a1=arima(vp,order=c(0,0,0))

a2=arima(vp,order=c(0,0,1))

a3=arima(vp,order=c(0,1,0))

a4=arima(vp,order=c(1,0,0))

a5=arima(vp,order=c(1,1,0))

a6=arima(vp,order=c(0,1,1))

a7=arima(vp,order=c(1,0,1))

a8=arima(vp,order=c(1,1,1))

a8=arima(vp,order=c(0,0,2))

a9=arima(vp,order=c(0,2,0))

a10=arima(vp,order=c(2,0,0))

veca=as.vector(cbind(a1$aic,a2$aic,a3$aic,a4$aic,

a5$aic,a6$aic,a7$aic,a8$aic,a9$aic,a10$aic))

which.min(veca)

> which.min(veca)

[1] 4

>

# Output of a4 model ....

> veca[4]

[1] -6234.531

> a4

Call:

arima(x = vp, order = c(1, 0, 0))

Coefficients:

ar1 intercept

0.4163 1.0054

s.e. 0.0287 0.0006

sigma^2 estimated as 0.0001148: log likelihood = 3120.27, aic = -6234.53

Sub-task - perform search for optimal arima using Monte Carlo method

a1) change in one parameter, task - create ormax dimensional vector

of aic values

d1<-decompose(log(UKDriverDeaths))

d1r<-d1$random

d1rr<-d1r[10:180]

ordmax<-20

aric<-rep(0,ordmax)

for(i1 in 0:ordmax){

ar<-arima(d1r,order=c(i1,0,0))

aric[i1]<-ar$aic

}

plot(aric)

ARIMA simulator

# compare and estimate role of autoregression

a) try to investigate influence of autoregressive term

ars<-arima.sim(n = 1000, list(ar = c(0.6897, -0.4858), 
ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

plot(acf(ars))

# change strength of autoregression term

ars1<-arima.sim(n = 1000, list(ar = c(0.9897, -0.4858), ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

acf0<-acf(ars)

acf1<-acf(ars1)

plot(acf0[[1]],col=4, type="l")

lines(acf1[[1]], col=2)

b) demonstrate influence of positive autoregressive term

ars1<-arima.sim(n = 2000, list(ar = c(0.1, -0.4858), ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

ars2<-arima.sim(n = 2000, list(ar = c(0.3, -0.4858), ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

ars3<-arima.sim(n = 2000, list(ar = c(0.5, -0.4858), ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

ars4<-arima.sim(n = 2000, list(ar = c(0.9, -0.4858), ma = c(-0.2279, 0.2488)),sd = sqrt(0.1796))

jpeg(filename = "acf14.jpg")

par(mfrow=c(2,2))

acf(ars1)

acf(ars2)

acf(ars3)

acf(ars4)

dev.off()

VAR model

library(vars)

data(Canada)

var.2c <- VAR(Canada, p = 2, type = "const")

> var.2c

VAR Estimation Results:

=======================

Estimated coefficients for equation e:

======================================

Call:

e = e.l1 + prod.l1 + rw.l1 + U.l1 + e.l2 + prod.l2 + rw.l2 + U.l2 + const

e.l1 prod.l1 rw.l1 U.l1 e.l2

1.637821e+00 1.672717e-01 -6.311863e-02 2.655848e-01 -4.971338e-01

prod.l2 rw.l2 U.l2 const

-1.016501e-01 3.844492e-03 1.326893e-01 -1.369984e+02

causality(var.2c, cause = "e")

> causality(var.2c, cause = "e")

$Granger

Granger causality H0: e do not Granger-cause prod rw U

data: VAR object var.2c

F-Test = 6.2768, df1 = 6, df2 = 292, p-value = 3.206e-06

$Instant

H0: No instantaneous causality between: e and prod rw U

data: VAR object var.2c

Chi-squared = 26.0685, df = 3, p-value = 9.228e-06

# use a robust HC variance-covariance matrix for the Granger test:

causality(var.2c, cause = "e", vcov.=vcovHC(var.2c))

http://multidimenstat.weblahko.sk/lecture_time_series1.pdf