Sovita AR(p) malli SP500 tuottojen itseisarvoihin.
file<-"http://cc.oulu.fi/~jklemela/timeseries/sp500.csv"
data<-read.csv(file=file)
sp500<-data[,7] # otetaan kunkin paivan lopetuskurssi
sp500<-sp500[length(sp500):1] # aloitetaan aikasarja vanhimmasta havainnosta
# piiretaan aikasarja
plot(sp500,type="l")
# tulostetaan tuotot
y<-sp500[2:length(sp500)]-sp500[1:(length(sp500)-1)]
r<-y/sp500[1:(length(sp500)-1)]
plot(r,type="l")
plot(abs(r),type="l")
library(tseries)
# sovitetaan AR-malli -> arma
p<-20
order<-c(p,0,0)
ari<-arma(abs(r),order=order)
ari
summary(ari)
Call:
arma(x = abs(r), order = order)
Coefficient(s):
ar1 ar2 ar3 ar4 ar5 ar6 ar7
0.060794 0.100058 0.064395 0.055392 0.109019 0.048397 0.044250
ar8 ar9 ar10 ar11 ar12 ar13 ar14
0.036990 0.038166 0.030609 0.052837 0.027897 0.015074 0.002718
ar15 ar16 ar17 ar18 ar19 ar20 intercept
0.020151 0.021892 0.020400 0.036885 0.015948 0.022011 0.001161
Call:
arma(x = abs(r), order = order)
Model:
ARMA(20,0)
Residuals:
Min 1Q Median 3Q Max
-0.040189 -0.003684 -0.001267 0.002467 0.190536
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
ar1 0.0607937 0.0078922 7.703 1.33e-14 ***
ar2 0.1000583 0.0079068 12.655 < 2e-16 ***
ar3 0.0643954 0.0079410 8.109 4.44e-16 ***
ar4 0.0553917 0.0079557 6.963 3.34e-12 ***
ar5 0.1090188 0.0079659 13.686 < 2e-16 ***
ar6 0.0483974 0.0080105 6.042 1.52e-09 ***
ar7 0.0442496 0.0080195 5.518 3.43e-08 ***
ar8 0.0369903 0.0080264 4.609 4.05e-06 ***
ar9 0.0381662 0.0080284 4.754 2.00e-06 ***
ar10 0.0306089 0.0080231 3.815 0.000136 ***
ar11 0.0528370 0.0080231 6.586 4.53e-11 ***
ar12 0.0278970 0.0080282 3.475 0.000511 ***
ar13 0.0150737 0.0080260 1.878 0.060366 .
ar14 0.0027177 0.0080180 0.339 0.734647
ar15 0.0201508 0.0080088 2.516 0.011867 *
ar16 0.0218917 0.0079643 2.749 0.005983 **
ar17 0.0204001 0.0079543 2.565 0.010328 *
ar18 0.0368855 0.0079398 4.646 3.39e-06 ***
ar19 0.0159482 0.0079061 2.017 0.043674 *
ar20 0.0220106 0.0078922 2.789 0.005289 **
intercept 0.0011609 0.0001012 11.467 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Fit:
sigma^2 estimated as 4.114e-05, Conditional Sum-of-Squares = 0.66, AIC = -116470.9
# residuaalit ################################
resi<-residuals(ari)
plot(resi)
acf(resi,na.action=na.pass)
# AIC -> arima0 ###############################
p<-30
aics<-matrix(0,p,1)
for (i in 1:p){
order<-c(i,0,0)
aics[i]<-arima0(r,order=order)$aic
}
plot(aics)
aics<-c(
-103145.1,
-103174.5,
-103172.5,
-103171.3,
-103173.5,
-103172.1,
-103177.2,
-103176.7,
-103176.1,
-103176.6,
-103178.5,
-103189.1,
-103187.1,
-103185.1,
-103185.1,
-103200.1,
-103199.0,
-103204.3,
-103202.4,
-103201.3,
-103207.2,
-103205.2,
-103203.3,
-103202.2,
-103202.4,
-103207.7,
-103211.9,
-103210.0,
-103217.6,
-103216.5
)
plot(aics,type="b")
p<-30
aics<-matrix(0,p,1)
for (i in 1:p){
order<-c(i,0,0)
aics[i]<-arima0(abs(r),order=order)$aic
}
plot(aics)
aics
aics<-c(
-113887.8,
-114749.5,
-115170.9,
-115444.7,
-115876.5,
-116013.9,
-116117.5,
-116188.3,
-116253.2,
-116295.5,
-116371.9,
-116398.2,
-116409.6,
-116411.0,
-116424.4,
-116436.7,
-116445.6,
-116469.6,
-116472.4,
-116478.0,
-116482.8,
-116481.0,
-116488.0,
-116494.3,
-116494.0,
-116492.2,
-116522.7,
-116523.7,
-116532.8
)
plot(aics)
# Sum of squared prediction errors ####################
p<-10
sspe<-matrix(0,p,1)
t0<-10000
T<-length(r)
for (i in 1:p){
order<-c(i,0)
errs<-matrix(0,T-t0,1)
for (t in t0:(T-1)){
x<-r[1:t]
coef<-arma(x,order=order)$coef
rhat<-sum(as.vector(coef[1:i])*r[t:(t-i+1)])
errs[t-t0+1]<-r[t+1]-rhat
}
sspe[i]<-sqrt(sum(errs^2))
}
plot(sspe)
sspe<-c(
0.9058649,
0.9058457,
0.9060230,
0.9061900,
0.9063012,
0.9066239,
0.9064767,
0.9065603,
0.9068165,
0.9068918
)
plot(sspe,type="b")