TianyuDu/AnnEconForecast

# Set data directory path
setwd("C:/Teaching/Undergrad/data ECO374")
 
# Install and load required packages
chooseCRANmirror(graphics=FALSE, ind=33)
if (!require("fGarch")) install.packages("fGarch")

library(aTSA)
library(xts)
library(anytime)
library(MASS)
library(moments)
library(fGarch)

# Simulate ARCH(p)
spec = garchSpec(model = list(omega = 2, alpha = c(0.3, 0.1, 0.4), beta = 0))
ARCHp <- garchSim(spec, n = 1000, extended = TRUE)

plot(ARCHp$garch, type="l", main = expression(paste("ARCH(3): ", alpha[1], " = 0.3,  ", alpha[2], " = 0.1,  ", alpha[3], " = 0.4")), cex.main=0.9)

# Distributions of ARCH draws
# Note that the distribution is not Normal in a similar way to a typical financial time series.
r_t <- ARCHp$garch
fit <- fitdistr(r_t, "normal")
para <- fit$estimate
hist(r_t, prob = TRUE, breaks=100, xlim=c(-11,11), col="lightgrey", main="Histogram of ARCH draws", cex.main=0.9)
curve(dnorm(x, para[1], para[2]), col = 2, add = TRUE)

# Moments of the ARCH draws (alpha=0.6)
all.moments(r_t, central=TRUE, order.max=4)

# ACF and PACF of the ARCH draws and their square (variance)
maxlag <- 30
ACF <- acf(r_t, main=NULL, lag.max=maxlag, plot=FALSE)
PACF <- pacf(r_t, main=NULL, lag.max=maxlag, plot=FALSE)

ACF_S <- acf(r_t^2, main=NULL, lag.max=maxlag, plot=FALSE)
PACF_S <- pacf(r_t^2, main=NULL, lag.max=maxlag, plot=FALSE)

par(mfcol=c(2,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(ACF[1:maxlag], ylim=c(-0.1,0.5), main="ACF of ARCH draws")
plot(PACF[1:maxlag], ylim=c(-0.1,0.5),main="PACF of ARCH draws")
plot(ACF_S[1:maxlag], ylim=c(-0.1,0.5), main="ACF of ARCH draws SQUARED")
plot(PACF_S[1:maxlag], ylim=c(-0.1,0.5),main="PACF of ARCH draws SQUARED")

# ACF and PACF of the normalized ARCH draws (variance) 
# Note that all the information of the process is contained in the conditional variance.
# Once we normalize with it, there is no more information left.
maxlag <- 20
r_t_normalized = ARCHp$garch / ARCHp$sigma
ACF <- acf(r_t_normalized, main=NULL, lag.max=maxlag, plot=FALSE)
PACF <- pacf(r_t_normalized, main=NULL, lag.max=maxlag, plot=FALSE)

r_t_S_normalized = ARCHp$garch^2 / ARCHp$sigma^2
ACF_S <- acf(r_t_S_normalized, main=NULL, lag.max=maxlag, plot=FALSE)
PACF_S <- pacf(r_t_S_normalized, main=NULL, lag.max=maxlag, plot=FALSE)

par(mfcol=c(2,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(ACF[1:maxlag], main="ACF of ARCH draws normalized")
plot(PACF[1:maxlag], main="PACF of ARCH draws normalized")

plot(ACF_S[1:maxlag], main="ACF of ARCH draws SQUARED normalized")
plot(PACF_S[1:maxlag], main="PACF of ARCH draws SQUARED normalized")