GARCH Models
- Normal day distribution/
Turbuolence day distribution
- Relative variation of the
series
- Mean
- Standard deviation (σ_t)
- Annual volatility [sqrt(252)*dayli_volatility
R packages
- Xts
- PerformanceAnalytics
- Rugarch
-
https://www.datacamp.com/cheat-sheet/xts-cheat-sheet-time-series-in-r
library(rugarch)
library(xts)
library(tydiverse)
# GARCH familiarization
alpha <- 0.1
beta <- 0.8
sp500ret <- rnorm(1000, mean = 0, sd = 0.8)
omega <- var(sp500ret) * (1 - alpha - beta)
omega
plot(sp500ret, type = "l")
e <- sp500ret-mean(sp500ret)
e2 <- e^2
plot(e2, type = "l")
# predicte variance
nobs <- length(sp500ret)
predvar <- rep(NA, nobs)
predvar[1] <- var(sp500ret)
for (t in 2:nobs){
predvar[t] <- omega + alpha*e2[t-1] + beta*predvar[t-1]
}
plot(predvar, type="l")
# GARCH volatility
# sqrt of predicted vairance
predvol <- sqrt(predvar)
predvol <- xts(predvol, order.by = time(sp500ret))
# plot(predvol, type = "l")
# unconditional volatility
uncvol <-sqrt(omega/(1-alpha-beta))
uncvol <-rep(uncvol, nobs)
uncovol <-xts((uncvol), order.by = time(sp500ret))
plot(predvol, type = "l")
lines(uncvol, col = "red", lwd =2)
# UGARCH package
garchspec <- ugarchspec(mean.model = list(armaOrder = c(0,0)),
variance.model = list(model = "sGARCH"),
distribution.model = "norm")
garchfit <- ugarchfit(data = sp500ret, spec = garchspec)
garchforecast <- ugarchforecast(fitORspec = garchfit, n.ahead = 5)
garchcoef <- coef(garchfit)
garchvol <- sigma(garchfit)
plot(garchvol)
tail(garchvol, 1)
sigma(garchforecast)
------------------------
Return.annualized() and StdDev.annualized()
skewness()
kurtosis().
qqnorm()
library(rugarch)
library(xts)
library(tydiverse)
# GARCH familiarization
alpha <- 0.1
beta <- 0.8
sp500ret <- rnorm(1000, mean = 0, sd = 0.8)
omega <- var(sp500ret) * (1 - alpha - beta)
omega
plot(sp500ret, type = "l")
e <- sp500ret-mean(sp500ret)
e2 <- e^2
plot(e2, type = "l")
# predicte variance
nobs <- length(sp500ret)
predvar <- rep(NA, nobs)
predvar[1] <- var(sp500ret)
for (t in 2:nobs){
predvar[t] <- omega + alpha*e2[t-1] + beta*predvar[t-1]
}
plot(predvar, type="l")
# GARCH volatility
# sqrt of predicted vairance
predvol <- sqrt(predvar)
predvol <- xts(predvol, order.by = time(sp500ret))
# plot(predvol, type = "l")
# unconditional volatility
uncvol <-sqrt(omega/(1-alpha-beta))
uncvol <-rep(uncvol, nobs)
uncovol <-xts((uncvol), order.by = time(sp500ret))
plot(predvol, type = "l")
lines(uncvol, col = "red", lwd =2)
# UGARCH package
garchspec <- ugarchspec(mean.model = list(armaOrder = c(0,0)),
variance.model = list(model = "sGARCH"),
distribution.model = "norm")
garchfit <- ugarchfit(data = sp500ret, spec = garchspec)
garchforecast <- ugarchforecast(fitORspec = garchfit, n.ahead = 5)
garchcoef <- coef(garchfit)
garchvol <- sigma(garchfit)
plot(garchvol)
tail(garchvol, 1)
sigma(garchforecast)
#//////////////////////////////////////////////////////////////////////////////-
#------------------------------------------------------------------------------/
#------------------------------------------------------------------------------/
# Normal simultation and testing for normality
par(mfrow=c(2,2))
serie1 <- rnorm(1000, mean=3, sd=2)
qqnorm(serie1)
qqline(serie1)
plot(serie1)
plot(serie1, type="l")
lines(rep(mean(serie1), length(serie1)), col="red")
hist(serie1, nclass=100, probability=TRUE)
lines(seq(min(serie1), max(serie1), lenght(serei1)), dnorm(serie1, mean=mean(serie1), sd=sd(serie1)), col="red")
hist(serie1, nclass=100, probability=TRUE, col="blue")
lines(density(serie1), col="red", lwd=2)
moments::skewness(serie1)
moments::kurtosis(serie1)
moments::jarque.test(serie1)
# Daily returns are usually very non-normal
# By CLT longer-intervasl (summarization) suggest they may by more normal
#------------------------------------------------------------------------------/
#------------------------------------------------------------------------------/
# t-student
# the parameter "v", as "v" gest largter the distribution tends to normal
# rstudent()
n <- 1000
df <- 30
t_values <- rt(n, df)
hist(t_values, breaks = 40, probability = TRUE,
main = paste("Student's t-distribution (df =", df, ")"),
xlab = "Value", col = "lightblue", border = "black")
lines(density(t_values), col = "red", lwd = 2)
tfit <- QRM::fit.st(t_values)
tpars <- tfit$par.ests
nu <- tpars[1]
mu <- tpars[2]
sigma <- tpars[3]
hist(t_values, nclass=100, probability=TRUE)
lines(t_values, dnorm(t_values, mean=mean(t_values), sd=sd(t_values)), col="red")
yvals <- dt((t_values-mu)/sigma, df=nu)/sigma
lines(t_values, yvals, col="blue")
https://www.datacamp.com/cheat-sheet/xts-cheat-sheet-time-series-in-r
