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## :=library(sjPlot)## Warning: package 'sjPlot' was built under R version 4.2.2#Setting the working directory
setwd("C:/Users/user/Pictures/BlueStacks/New folder")#Data import
AEX<- read_csv("AEX.csv")
AEX['Date'] <- as.Date(AEX$Date)
#selecting the 'close' and date columns of data set
AEXprice<-AEX$Close
AEXdate<-AEX$Date
AEX.TS<- xts(AEXprice,order.by=AEXdate)#log returns of the data
logAEX.rtn<-diff(log(AEXprice), lag=1)#Normality Tests
#JB Test
jb5 <- jarqueberaTest(logAEX.rtn);jb5##
## Title:
## Jarque - Bera Normalality Test
##
## Test Results:
## STATISTIC:
## X-squared: 13016.4897
## P VALUE:
## Asymptotic p Value: < 2.2e-16
##
## Description:
## Mon Jan 9 11:52:19 2023 by user: Nick Mutembei#ADF test
adf.test(logAEX.rtn)## Augmented Dickey-Fuller Test
## alternative: stationary
##
## Type 1: no drift no trend
## lag ADF p.value
## [1,] 0 -37.7 0.01
## [2,] 1 -26.5 0.01
## [3,] 2 -23.5 0.01
## [4,] 3 -19.7 0.01
## [5,] 4 -17.2 0.01
## [6,] 5 -16.3 0.01
## [7,] 6 -15.0 0.01
## [8,] 7 -14.2 0.01
## Type 2: with drift no trend
## lag ADF p.value
## [1,] 0 -37.7 0.01
## [2,] 1 -26.5 0.01
## [3,] 2 -23.5 0.01
## [4,] 3 -19.7 0.01
## [5,] 4 -17.2 0.01
## [6,] 5 -16.4 0.01
## [7,] 6 -15.0 0.01
## [8,] 7 -14.2 0.01
## Type 3: with drift and trend
## lag ADF p.value
## [1,] 0 -37.7 0.01
## [2,] 1 -26.6 0.01
## [3,] 2 -23.6 0.01
## [4,] 3 -19.8 0.01
## [5,] 4 -17.3 0.01
## [6,] 5 -16.5 0.01
## [7,] 6 -15.1 0.01
## [8,] 7 -14.3 0.01
## ----
## Note: in fact, p.value = 0.01 means p.value <= 0.01#Normality Test
#histogram
#Is the mean log return significantly different from zero=T test
t.test(logAEX.rtn)##
## One Sample t-test
##
## data: logAEX.rtn
## t = -0.53679, df = 1380, p-value = 0.5915
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## -0.003412969 0.001946428
## sample estimates:
## mean of x
## -0.0007332706#Normality Tests#Plotting the average daily returns ofdataset
p1 <- ggplot(AEX.TS, aes(x=AEXdate,y=AEXprice))+geom_line()+
theme(plot.title=element_text(hjust=0.5))+
labs(x="Date",y="Price",title="Average Daily of the Stock Indices")
p1
#stationarity Test
ggplot(AEX.TS[-1], aes(AEXdate[-1],logAEX.rtn))+geom_line()+
labs(x="Date",y="Log Return",title = " Log Returns plot")
# Test set 500 observations
specgarchaex = ugarchspec(variance.model = list(garchOrder=c(1,1)), distribution.model ="sstd",mean.model =list(armaOrder=c(0,0,2)) )
roll = ugarchroll(specgarchaex, data = logAEX.rtn, n.ahead = 1, forecast.length = 500,refit.every = 25, refit.window = "rolling",
parallel = TRUE,parallel.control = list(pkg = "snowfall", cores = 10), solver = "solnp",
solver.control = list(tol = 1e-05, delta = 1e-06, trace = 0),calculate.VaR = TRUE, VaR.alpha = c(0.01, 0.025, 0.05))
report(roll, type = "VaR", VaR.alpha = 0.01, conf.level = 0.95)## VaR Backtest Report
## ===========================================
## Model: sGARCH-sstd
## Backtest Length: 500
## Data:
##
## ==========================================
## alpha: 1%
## Expected Exceed: 5
## Actual VaR Exceed: 11
## Actual %: 2.2%
##
## Unconditional Coverage (Kupiec)
## Null-Hypothesis: Correct Exceedances
## LR.uc Statistic: 5.419
## LR.uc Critical: 3.841
## LR.uc p-value: 0.02
## Reject Null: YES
##
## Conditional Coverage (Christoffersen)
## Null-Hypothesis: Correct Exceedances and
## Independence of Failures
## LR.cc Statistic: 6.848
## LR.cc Critical: 5.991
## LR.cc p-value: 0.033
## Reject Null: YESreport(roll,type="VaR",VaR.alpha=0.05,conf.level=0.95)## VaR Backtest Report
## ===========================================
## Model: sGARCH-sstd
## Backtest Length: 500
## Data:
##
## ==========================================
## alpha: 5%
## Expected Exceed: 25
## Actual VaR Exceed: 31
## Actual %: 6.2%
##
## Unconditional Coverage (Kupiec)
## Null-Hypothesis: Correct Exceedances
## LR.uc Statistic: 1.413
## LR.uc Critical: 3.841
## LR.uc p-value: 0.235
## Reject Null: NO
##
## Conditional Coverage (Christoffersen)
## Null-Hypothesis: Correct Exceedances and
## Independence of Failures
## LR.cc Statistic: 5.501
## LR.cc Critical: 5.991
## LR.cc p-value: 0.064
## Reject Null: NOfit= ugarchfit(specgarchaex,data=logAEX.rtn[1:965,drop =
FALSE])
Spec=specgarchaex;setfixed(Spec)<-as.list(coef(fit))
filt = ugarchfilter(Spec, logAEX.rtn[965:1381, drop = FALSE]);
actual = logAEX.rtn[965:1381]
VaR = fitted(filt) + sigma(filt)*qdist("sstd", p=0.05, mu=0,
sigma=1,skew=coef(fit)["skew"], shape=coef(fit)["shape"])
f = function(x) qdist("sstd",p=x,mu=0,sigma=1,skew=
coef(fit)["skew"], shape=coef(fit)["shape"])
ES = fitted(filt) + sigma(filt)*integrate(f, 0, 0.01) $value/0.01;print(ESTest(0.01, actual,ES, VaR, boot = T))## $expected.exceed
## [1] 4
##
## $actual.exceed
## [1] 28
##
## $H1
## [1] "Mean of Excess Violations of VaR is greater than zero"
##
## $boot.p.value
## [1] 1
##
## $p.value
## [1] 1
##
## $Decision
## [1] "Fail to Reject H0"#back tetsing
p = c()
p[1] = pbinom(q = 0 , size = 500 , prob = 0.05)
for(i in 1:50){
p[i] = (pbinom(q = (i-1) , size = 500 , prob = 0.05) - pbinom(q = (i-2) , size = 500 , prob = 0.05))
}
VaR95_td = mean(logAEX.rtn) + roll@forecast$density[,'Sigma']*qdist(distribution='std', shape=3.7545967917, p=0.05)qplot(y = VaR95_td , x = 1:500 , geom = 'line') +
geom_point(aes(x = 1:500 , y = logAEX.rtn[759:1258] , color = as.factor(logAEX.rtn[759:1258] < VaR95_td)) , size = 2) + scale_color_manual(values = c('gray' , 'red')) +
labs(y = 'Daily Returns' , x = 'Test set Observation') + theme_light() +
theme(legend.position = 'none')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
# VaR and ES at 0.01
#AEX
ys = sort(logAEX.rtn)
op = length(logAEX.rtn)*0.01
VaR = -ys[op]*500
print(VaR)## [1] 73.14607ES = -mean(ys[1:op])*500
print(ES)## [1] 94.29972