library("quantmod")
## Loading required package: xts
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
## Loading required package: TTR
## Version 0.4-0 included new data defaults. See ?getSymbols.
da=read.table("q-jnj-earns-9211.txt",header = T)
head(da)
## day mon year earns
## 1 30 1 1992 0.11
## 2 23 4 1992 0.18
## 3 21 7 1992 0.18
## 4 20 10 1992 0.17
## 5 1 2 1993 0.12
## 6 29 4 1993 0.20
jnj=da$earns
jnj1=ts(jnj,frequency=12,start=c(1992,1))
par(mfcol=c(2,1))
plot(jnj1,xlab="year",ylab="returns")
title(main="(a): Simple returns")
acf(jnj,lag=24) # command to obtain sample ACF of the data
ln.jnj=log(jnj+1)
Box.test(jnj,lag=12,type="Ljung")
##
## Box-Ljung test
##
## data: jnj
## X-squared = 584.59, df = 12, p-value < 2.2e-16
Box.test(ln.jnj,lag=12,type="Ljung")
##
## Box-Ljung test
##
## data: ln.jnj
## X-squared = 598.48, df = 12, p-value < 2.2e-16
jnj2=ts(jnj,frequency=12,start=c(1992,1))
par(mfcol=c(2,1))
plot(jnj1,xlab="year",ylab="returns")
title(main="(a): Simple returns")
acf(jnj1,lag=24)
gnp=diff(ln.jnj)
dim(da)
## [1] 78 4
tdx=c(1:78)/4+1992
par(mfcol=c(2,1))
plot(tdx,ln.jnj,xlab="year",ylab="gnp",type="l")
plot(tdx[2:78],gnp,type="l",xlab="year",ylab="growth")
acf(gnp,lag=12)
pacf(gnp,lag=12) # compute PACF
m1=arima(gnp,order=c(3,0,0))
m1
##
## Call:
## arima(x = gnp, order = c(3, 0, 0))
##
## Coefficients:
## ar1 ar2 ar3 intercept
## -0.9547 -0.9342 -0.9461 0.0087
## s.e. 0.0324 0.0363 0.0277 0.0006
##
## sigma^2 estimated as 0.0003623: log likelihood = 192.15, aic = -374.3
tsdiag(m1,gof=12) # model checking discussed later
p1=c(1,-m1$coef[1:3]) # set-up the polynomial
r1=polyroot(p1) # solve the polynomial equation
r1
## [1] 0.014824+1.019333i -1.017089+0.000000i 0.014824-1.019333i
Mod(r1)
## [1] 1.019441 1.017089 1.019441
k=2*pi/acos(0.014824/1.019441)
k
## [1] 4.037376
mm1=ar(gnp,method="mle")
mm1$order
## [1] 4
names(mm1)
## [1] "order" "ar" "var.pred" "x.mean"
## [5] "aic" "n.used" "n.obs" "order.max"
## [9] "partialacf" "resid" "method" "series"
## [13] "frequency" "call" "asy.var.coef"
print(mm1$aic,digits=3)
## 0 1 2 3 4 5 6 7 8
## 251.685 232.820 226.296 51.073 0.000 1.821 3.812 5.118 0.102
## 9 10 11 12
## 1.430 1.319 2.384 4.215
aic=mm1$aic
length(aic)
## [1] 13
plot(c(0:12),aic,type="h",xlab="order",ylab="aic")
lines(0:12,aic,lty=2)
#Forecasting
m1=arima(jnj,order=c(0,0,9))
m1
##
## Call:
## arima(x = jnj, order = c(0, 0, 9))
##
## Coefficients:
## ma1 ma2 ma3 ma4 ma5 ma6 ma7 ma8
## 1.0058 1.2457 1.2014 2.1391 1.5133 1.319 1.0087 1.2378
## s.e. 0.1272 0.1732 0.2721 0.2876 0.2363 0.215 0.2427 0.2446
## ma9 intercept
## 0.3804 0.6094
## s.e. 0.1330 0.0791
##
## sigma^2 estimated as 0.003708: log likelihood = 96.24, aic = -170.47
m1=arima(jnj,order=c(0,0,9),fixed=c(NA,0,NA,0,0,0,0,0,NA,NA))
m1
##
## Call:
## arima(x = jnj, order = c(0, 0, 9), fixed = c(NA, 0, NA, 0, 0, 0, 0, 0, NA, NA))
##
## Coefficients:
## ma1 ma2 ma3 ma4 ma5 ma6 ma7 ma8 ma9 intercept
## 0.1294 0 0.6503 0 0 0 0 0 -0.2064 0.6002
## s.e. 0.2891 0 0.3004 0 0 0 0 0 0.2011 0.0468
##
## sigma^2 estimated as 0.06913: log likelihood = -7.43, aic = 24.87
sqrt(0.06913)
## [1] 0.2629258
Box.test(m1$residuals,lag=12,type="Ljung")
##
## Box-Ljung test
##
## data: m1$residuals
## X-squared = 394.68, df = 12, p-value < 2.2e-16
pv=1-pchisq(394.68,9)
pv
## [1] 0
m1=arima(jnj[1:78],order=c(0,0,9),fixed=c(NA,0,NA,0,0,0,0,0,NA,NA))
m1
##
## Call:
## arima(x = jnj[1:78], order = c(0, 0, 9), fixed = c(NA, 0, NA, 0, 0, 0, 0, 0,
## NA, NA))
##
## Coefficients:
## ma1 ma2 ma3 ma4 ma5 ma6 ma7 ma8 ma9 intercept
## 0.1294 0 0.6503 0 0 0 0 0 -0.2064 0.6002
## s.e. 0.2891 0 0.3004 0 0 0 0 0 0.2011 0.0468
##
## sigma^2 estimated as 0.06913: log likelihood = -7.43, aic = 24.87
predict(m1,10)
## $pred
## Time Series:
## Start = 79
## End = 88
## Frequency = 1
## [1] 0.9486664 0.6352115 0.8469624 0.5757401 0.4878375 0.5478441 0.4812723
## [8] 0.5757913 0.4822479 0.6002087
##
## $se
## Time Series:
## Start = 79
## End = 88
## Frequency = 1
## [1] 0.2629332 0.2651269 0.2651269 0.3154777 0.3154777 0.3154777 0.3154777
## [8] 0.3154777 0.3154777 0.3201099