Sovita GARCH(1,1) malli SP500 tuottoihin, piirra hat(sigma)_t ja residuaalit.
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")
# Estimointi GARCH(1,1) #######################################
library(tseries)
x<-100*r
order<-c(1,1)
ga<-garch(x,order=order)
summary(ga)
Call:
garch(x = x, order = order)
Model:
GARCH(1,1)
Residuals:
Min 1Q Median 3Q Max
-10.65981 -0.52453 0.06144 0.64030 6.44386
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 0.0074141 0.0006603 11.23 <2e-16 ***
a1 0.0775141 0.0016973 45.67 <2e-16 ***
b1 0.9164638 0.0022025 416.10 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 9478.852, df = 2, p-value < 2.2e-16
Box-Ljung test
data: Squared.Residuals
X-squared = 7.7792, df = 1, p-value = 0.005285
# Estimointi GARCH(1,3) ###################################
# 'order[2]' corresponds to the ARCH part and 'order[1]’ to the GARCH part
# GARCH(p,q): p on ARCH-part ja q on GARCH-part
x<-100*r
order<-c(3,1)
ga<-garch(x,order=order)
summary(ga)
Call:
garch(x = x, order = order)
Model:
GARCH(3,1)
Residuals:
Min 1Q Median 3Q Max
-10.60827 -0.52684 0.06149 0.63883 6.52770
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 9.410e-03 9.158e-04 10.275 < 2e-16 ***
a1 1.055e-01 3.925e-03 26.881 < 2e-16 ***
b1 6.520e-01 5.869e-02 11.110 < 2e-16 ***
b2 4.893e-13 8.861e-02 0.000 1
b3 2.350e-01 5.634e-02 4.171 3.03e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 8724.943, df = 2, p-value < 2.2e-16
Box-Ljung test
data: Squared.Residuals
X-squared = 1.0553, df = 1, p-value = 0.3043
# Fan & Yao, 1974-1999 s. 174: #################################
# a0 0.015
# a1 0.112
# b1 0.492
# b2 -0.034
# b3 0.420
# hat(sigma)_t ######################################
a0<-0.0074141
a1<-0.0775141
b1<-0.9164638
sig<-sd(x) # [1] 0.9729996
sig<-sqrt(a0/(1-a1-b1)) # [1] 1.109571
T<-length(x)
hatsigma2<-matrix(sig^2,T,1)
for (i in 2:T) hatsigma2[i]<-a0+a1*x[i-1]^2+b1*hatsigma2[i-1]
plot(sqrt(hatsigma2),type="l")
# residuaalit #######################################
hateps<-matrix(0,T,1)
for (i in 1:T) hateps[i]<-x[i]/sqrt(hatsigma2[i])
plot(hateps,type="l")
# Akaiken informaatiokriteeri ##########################
AIC(ga)
# [1] 38653.25
aics<-matrix(0,4,1)
for (i in 1:4){
order<-c(1,i)
ga<-garch(r,order=order)
aics[i]<-AIC(ga)
}
plot(aics)
aics
[,1]
[1,] -106356.3
[2,] -106292.0
[3,] -106136.5
[4,] -105892.5
##########################
library(fSeries)
#######################################################
# kokeillaan aikaa 1972-2000, vrt. s.171, GARCH(1,3)
time<-data[,1]
time<-time[length(time):1]
time[1]
time[length(time)]
# [1] 1950-01-03
# [1] 2013-10-11
t0<-5500
t1<-12550
time[t0]
time[t1]
x<-100*r[t0:t1]
order<-c(3,1)
ga<-garch(x,order=order)
summary(ga)
Call:
garch(x = x, order = order)
Model:
GARCH(3,1)
Residuals:
Min 1Q Median 3Q Max
-10.11524 -0.53561 0.04789 0.64116 5.50138
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 1.221e-02 1.879e-03 6.497 8.18e-11 ***
a1 9.651e-02 3.730e-03 25.878 < 2e-16 ***
b1 4.718e-01 7.539e-02 6.258 3.90e-10 ***
b2 5.700e-14 9.047e-02 0.000 1
b3 4.205e-01 6.333e-02 6.640 3.14e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 4537.4, df = 2, p-value < 2.2e-16
Box-Ljung test
data: Squared.Residuals
X-squared = 1.8779, df = 1, p-value = 0.1706
##### hat(sigma)_t
a0<-1.221e-02 # 0.015
a1<-9.651e-02 # 0.112
b1<-4.718e-01 # 0.492
b2<-3.094e-14 #-0.034
b3<-4.205e-01 # 0.420
sig<-sd(x)
T<-length(x)
hatsig2<-matrix(sig^2,T,1)
for (i in 4:T) hatsig2[i]<-a0+a1*x[i-1]^2+b1*hatsig2[i-1]+b2*hatsig2[i-2]+b3*hatsig2[i-3]
plot(sqrt(hatsig2),type="l")
#### residuaalit
hateps<-matrix(0,T,1)
for (i in 1:T) hateps[i]<-x[i]/sqrt(hatsigma2[i])
plot(hateps,type="l")
plot(hateps,type="l")