ARCH GARCH model – Econistics.com

rm(list=ls())
library(FinTS) # For Arch Test
library(rugarch) # For Garch Models
library(tseries) # For unit root test
library(dynlm) # For using lags in the model
library(vars) # For using VAR
library(nlWaldTest) # For testing non-linear Wald Test
library(lmtest) # For BP Test
library(broom) # For table presentations
library(car) # For robust standard errors
library(sandwich)
library(knitr)
library(forecast)
library(ggplot2)
library(pdfetch) #for importing financial data
library(tsbox)
library(stats)
library(zoo)
library(vrtest)

Setting Theme

theme_set(theme_bw())

Collecting data

df <- pdfetch_YAHOO(fields = “close”, c(“^gspc”))

Convert to time-series objects

dfts <- ts(df, start = c(2007,3), end= c(2020,161), frequency = 365)
data.frame(dfts)

Visuals

plot.ts(dfts)
title(“Time Series Plot of GSPC”)

hist(dfts, main = “Histogram of GSPC”, freq = FALSE, col = “grey”)

plot.ts(diff(dfts), ylab= “Change in GSPC”)
title (“First difference of GSPC”)

Step 1: Normality Test and Log Transformation

shapiro.test(dfts)
df1 <- log(df) df <- df %>% mutate(logseries = log(“^gspc”))

Step 2: Check for Stationarity of variance and mean

plot.ts(df1)
title(“Time Series Plot of Log of GSPC”)

Auto.VR(df1) # Variance is not constant

adf.test(df1$’^gspc’, k = 3)
df2 <- diff(df1)
df2 <- na.remove(df2)
adf.test(df2, k = 2) # series is I(1)

Step 3: Determining AR and MA components

acf.df2 <- acf(df2, main = “ACF of GSPC”, lag.max = 50)
pacf.df2 <- pacf(df2, main = “PACF of GSPC”, lag.max = 50)

ARIMA (p,I,q) = (1, 1, 2)

Step 4: Estimating ARIMA model

auto.arima(df2)

ARIMA = (0,1,1)

arima011 <- arima(df1, order = c(0,1,1))
summary(arima011)
tsdiag(arima011)
checkresiduals(arima011)
autoplot(arima011)

arima212 <- arima(df1, order = c(2,1,2))
autoplot(arima212)

arimar <- arima011$residuals
ggtsdisplay(arimar, main = “GSP ARIMA Residuals”)

Step 5: Estimate mean equation r = beta + error

gsp.mean <- dynlm(arimar ~ 1)
summary(gsp.mean)

Step 6: Determination of ARCH Effect

ehatsq <- ts(resid(gsp.mean)^2)
gsp.archef <- dynlm(ehatsq ~ L(ehatsq))
summary(gsp.archef)

t <- nobs(gsp.mean)
q <- length(coef(gsp.archef))-1
rsq <- glance(gsp.archef)[[1]]
lm <- (t-q)*rsq
alpha <- 0.05
chicr <- qchisq(1-alpha, q)
lm
chicr

gsp.archef1 <- ArchTest(arimar, lags=1, demean = TRUE)
gsp.archef1

step 7 : Estimating arch garch equation

gsp.arch <- garch(arimar,c(1,3), control = garch.control(maxiter = 500, grad = “numerical”)) #first item in c() is garch effect and second is arch effect

sgsparch <- summary(gsp.arch)
sgsparch

hhat <- ts(2*gsp.arch$fitted.values[-1,1]^2)
plot.ts(hhat)
title(“Volatility of GSP”)

vol <- gsp.arch$fitted.values
vol