ARI Y -2020- From Discrete To Continuous GARCH Volatility Modelling with R

FROM DISCRETE TO CONTINUOUS GARCH VOLATILITY MODELLING WITH R

To cite this work (Çalışmaya atıfta bulunmak için)

ARI, Y. (2020). From Discrete to Continuous: GARCH Volatility Modelling with R. Retrieved from https://math-stat.net/garch-and-cogarch-modelling.htm

FROM DISCRETE TO CONTINUOUS:

GARCH VOLATILITY MODELLING WITH R

Yakup ARI, PhD

Department of Economics

Alanya Alaaddin Keykubat University, Turkey

https://orcid.org/0000-0002-5666-5365

yakup.ari@alanya.edu.tr

Abstract

The aim of this study is to arrange the R codes that facilitate the comparison of the simple GARCH model and its extensions. The outputs of each discrete GARCH model are prepared as matrices. In addition, continuous GARCH model codes are also given. For this purpose, data of the platinium prices were used.The Platinium prices (PL=F) data is downloaded from “finance.yahoo.com” via “quantmod” package (Ryan et al., 2018) in R. The discrete GARCH family models and COGARCH model are fitted to returns of the PL=F using the “rugarch” package (Ghalanos, 2020a, 2020b) and later a COGARCH(1,1) by R package “yuima” by Iacus et al. (2017, 2018).

Keywords: GARCH Comparison, COGARCH, R Project, rugarch, yuima

Some References

Ghalanos A. (2020a). rugarch:Univariate GARCH models.R package version 1.4-2. Available at:

https://cran.r-project.org/web/packages/rugarch/rugarch.pdf

Ghalanos A. (2020b). Introduction to the rugarch package.Technical Report Available at:

https://cran.r-project.org/web/packages/rugarch/vignettes/ Introduction_to_the_rugarch_package. pdf

Ryan, J.A. and Ulrich, J. M. (2018). quantmod: Quantitative Financial Modelling Framework.

R package version 0.4-13.https://CRAN.R-project.org/package=quantmod

Iacus SM, Mercuri L, Rroji E (2017). “COGARCH$(p, q)$: Simulation and Inference with the yuima Package.” Journal of Statistical Software,80(4), 1–49. doi: 10.18637/jss.v080.i04.

Iacus S. M. &Yoshida N. (2018). “Simulation and Inference for Stochastic Processes with YUIMA: A Comprehensive R Framework for SDEs and other Stochastic Processes”, Springer International Publishing, pp. 254-256.

##################### PART 1- LIBRARIES ####################

library(MCMCpack); library(rugarch); library(fGarch);
library(tseries); library(aTSA); library(yuima);
library(quantmod);

######################################################

##################### PART 2 - DATA ####################

#part 2.1 upload data by quantmod from yahoo-finance

getSymbols("PL=F", from="2018-01-01", to="2020-05-01")

S<-`PL=F`$`PL=F.Close`

X<- dailyReturn(`PL=F`)

# part 2.2 merging and omitting the na values

md<-merge(S,X)

na.md<-na.omit(md)

# part 2.3 descriptive stats and plots

stat.desc(S); describe(S); stat.desc(X); describe(X); length(X)

#Set your language

Sys.setlocale("LC_TIME", "English")

par(mfrow=c(2,1))

plot(X, main="Returns of PL=F")

plot(S, main="PL=F " )

##############################################################

############ PART 3 - STATIONARITY and ARCH EFFECT ###############

#unit root tests ADF and PP

adft.out = adf.test(X)

ppt.out = pp.test(X)

kpss.out = kpss.out(X)

mod<-estimate(X,p=0)

mod<-arima(X)

arch.test.out<-arch.test(mod)

arch.test.out

ljung.box.test.out<-Box.test (X, lag = 12, type = "Ljung")

ljung.box.test.out

garch.data = X

########################################################

#COMPARING GARCH - FAMILY MODELS

#VIA INFORMATION CRITERIA AND LOGLIKELIHOOD

########################################################

getSymbols("PL=F", from="2018-01-01", to="2020-05-01")

S<-`PL=F`$`PL=F.Close`

X<- dailyReturn(`PL=F`)
garch.data<-X

# Define a function for GARCH model specification and fitting
fit_garch_model <- function(model_type, garch_order, submodel, distribution, data) {
spec <- ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
variance.model = list(model = model_type, garchOrder = garch_order, submodel = submodel),
distribution.model = distribution)
return(ugarchfit(spec, data = data))
}

# Define the model types, orders, submodels, and distributions
model_types <- c("fGARCH", "eGARCH", "iGARCH") # Add other models as needed
garch_orders <- list(c(1, 1))
submodels <- c("GARCH", "AVGARCH", "GJRGARCH", "APARCH", "ALLGARCH", "NGARCH", "NAGARCH", "mcsGARCH") # Add others as needed
distributions <- c("norm", "snorm", "std", "sstd", "ged", "sged", "nig", "jsu")

# Loop over model types, orders, submodels, and distributions
results <- list()
for (model_type in model_types) {
for (order in garch_orders) {
for (submodel in submodels) {
for (distribution in distributions) {
model_name <- paste(model_type, submodel, distribution, sep = "_")
results[[model_name]] <- fit_garch_model(model_type, order, submodel, distribution, garch.data)
}
}
}
}

# Extract information criteria and likelihood
info_criteria <- lapply(results, infocriteria)
likelihoods <- lapply(results, likelihood)

########################################################

#COMPARING GARCH MODEL WITH VARIOUS DISTRIBUTIONS

#VIA INFORMATION CRITERIA AND LOGLIKELIHOOD

# VIA MSE, RMSE, MAD

########################################################

# Define a function to create GARCH specifications
create_garch_spec <- function(distribution) {
ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
variance.model = list(model = "fGARCH", garchOrder = c(1, 1), submodel = "GARCH"),
distribution.model = distribution)
}

# List of distributions
distributions <- c("norm", "snorm", "std", "sstd", "ged", "sged", "nig", "jsu")

# Initialize lists to store results
specs <- list()
fits <- list()
info_criteria <- list()
likelihoods <- list()

# Loop over distributions
for (dist in distributions) {
spec_name <- paste("spec.garch", dist, "11", sep = ".")
fit_name <- paste("fit.garch", dist, "11", sep = ".")

specs[[spec_name]] <- create_garch_spec(dist)
fits[[fit_name]] <- ugarchfit(specs[[spec_name]], data = garch.data)

info_criteria[[fit_name]] <- infocriteria(fits[[fit_name]])
likelihoods[[fit_name]] <- likelihood(fits[[fit_name]])
}

# Create matrix for GARCH11 results
matrix.garch11 <- matrix(0, length(distributions), 5,
dimnames = list(paste(distributions, "GARCH11", sep = " "),
c("Akaike", "Bayes", "Shibata", "Hannan-Quinn", "Likelihood")))

# Populate the matrix
for (i in 1:length(distributions)) {
fit_name <- paste("fit.garch", distributions[i], "11", sep = ".")
matrix.garch11[i, 1:4] <- info_criteria[[fit_name]][1, ]
matrix.garch11[i, 5] <- likelihoods[[fit_name]]
}

# Assuming you have already fitted the GARCH models as per the previous code

# Function to calculate forecast metrics
calculate_forecast_metrics <- function(actual, forecast) {
mse <- mean((actual - forecast)^2)
rmse <- sqrt(mse)
mad <- mean(abs(actual - forecast))
return(c(MSE = mse, RMSE = rmse, MAD = mad))
}

# Number of periods to forecast
n_forecast <- 10 # Adjust as needed

# Initialize lists to store forecasts and metrics
forecasts <- list()
forecast_metrics <- list()

# Loop over the fitted models to perform forecasting and calculate metrics
for (dist in distributions) {
fit_name <- paste("fit.garch", dist, "11", sep = ".")

# Forecasting
forecast <- ugarchforecast(fits[[fit_name]], n.ahead = n_forecast)
forecasts[[fit_name]] <- forecast

# Assuming 'garch.data' is the actual data and 'n_forecast' is the number of periods you want to forecast
# You need to adjust the actual data extraction based on your data structure
actual_data <- tail(garch.data, n_forecast)
forecasted_data <- as.numeric(forecast@forecast$seriesFor)

# Calculate metrics
forecast_metrics[[fit_name]] <- calculate_forecast_metrics(actual_data, forecasted_data)
}

# Display forecast metrics for each model
forecast_metrics

# Continuing from the previous code

# Initialize a matrix to store forecast metrics for each model
metrics_matrix <- matrix(0, length(distributions), 3,
dimnames = list(paste(distributions, "GARCH11", sep = " "),
c("MSE", "RMSE", "MAD")))

# Populate the matrix with forecast metrics
for (i in 1:length(distributions)) {
fit_name <- paste("fit.garch", distributions[i], "11", sep = ".")
metrics_matrix[i, ] <- forecast_metrics[[fit_name]]
}

# Display the matrix
metrics_matrix

# Function to calculate forecast metrics with heteroskedasticity adjustments
calculate_forecast_metrics <- function(actual, forecast, conditional_variance) {
errors <- actual - forecast
standardized_errors <- errors / sqrt(conditional_variance)

mse <- mean(errors^2)
rmse <- sqrt(mse)
mad <- mean(abs(errors))

# Heteroskedastic-adjusted metrics
hrmse <- sqrt(mean(standardized_errors^2))
hmae <- mean(abs(standardized_errors))

return(c(MSE = mse, RMSE = rmse, MAD = mad, HRMSE = hrmse, HMAE = hmae))
}

# Initialize a matrix to store forecast metrics for each model
metrics_matrix <- matrix(0, length(distributions), 5,
dimnames = list(paste(distributions, "GARCH11", sep = " "),
c("MSE", "RMSE", "MAD", "HRMSE", "HMAE")))

# Populate the matrix with forecast metrics
for (i in 1:length(distributions)) {
fit_name <- paste("fit.garch", distributions[i], "11", sep = ".")

# Extract conditional variance from the fitted model
conditional_variance <- sigma(fits[[fit_name]])^2

# Adjust the length of conditional variance to match the forecast length
conditional_variance <- tail(conditional_variance, n_forecast)

# Calculate metrics
forecast_metrics[[fit_name]] <- calculate_forecast_metrics(actual_data, forecasts[[fit_name]]@forecast$seriesFor, conditional_variance)

# Populate the matrix
metrics_matrix[i, ] <- forecast_metrics[[fit_name]]
}

# Display the matrix
metrics_matrix

########################################################

#COMPARING GARCH MODEL WITH VARIOUS DISTRIBUTIONS

# VIA MSE, RMSE, MAD, HRMSE, HMAE

########################################################

# Load necessary library
library(rugarch)

# Function to create GARCH specifications
create_garch_spec <- function(distribution) {
ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
variance.model = list(model = "fGARCH", garchOrder = c(1, 1), submodel = "GARCH"),
distribution.model = distribution)
}

# Function to calculate forecast metrics with heteroskedasticity adjustments
calculate_forecast_metrics <- function(actual, forecast, conditional_variance) {
errors <- actual - forecast
standardized_errors <- errors / sqrt(conditional_variance)

mse <- mean(errors^2)
rmse <- sqrt(mse)
mad <- mean(abs(errors))

# Heteroskedastic-adjusted metrics
hrmse <- sqrt(mean(standardized_errors^2))
hmae <- mean(abs(standardized_errors))

return(c(MSE = mse, RMSE = rmse, MAD = mad, HRMSE = hrmse, HMAE = hmae))
}

# List of distributions
distributions <- c("norm", "snorm", "std", "sstd", "ged", "sged", "nig", "jsu")

# Initialize lists to store results
specs <- list()
fits <- list()
forecasts <- list()
forecast_metrics <- list()

# Loop over distributions
for (dist in distributions) {
spec_name <- paste("spec.garch", dist, "11", sep = ".")
fit_name <- paste("fit.garch", dist, "11", sep = ".")

# Create and fit GARCH model
specs[[spec_name]] <- create_garch_spec(dist)
fits[[fit_name]] <- ugarchfit(specs[[spec_name]], data = garch.data)

# Forecasting
forecasts[[fit_name]] <- ugarchforecast(fits[[fit_name]], n.ahead = 10) # Adjust n.ahead as needed

# Calculate metrics
actual_data <- tail(garch.data, 10) # Adjust to match forecast length
forecasted_data <- as.numeric(forecasts[[fit_name]]@forecast$seriesFor)
conditional_variance <- tail(sigma(fits[[fit_name]])^2, 10) # Adjust to match forecast length
forecast_metrics[[fit_name]] <- calculate_forecast_metrics(actual_data, forecasted_data, conditional_variance)
}

# Initialize a matrix to store forecast metrics for each model
metrics_matrix <- matrix(0, length(distributions), 5,
dimnames = list(paste(distributions, "GARCH11", sep = " "),
c("MSE", "RMSE", "MAD", "HRMSE", "HMAE")))

# Populate the matrix with forecast metrics
for (i in 1:length(distributions)) {
fit_name <- paste("fit.garch", distributions[i], "11", sep = ".")
metrics_matrix[i, ] <- forecast_metrics[[fit_name]]
}

# Display the matrix
metrics_matrix

########################################################

#COMPARING STANDART GARCH MODEL

#VIA INFORMATION CRITERIA AND LOGLIKELIHOOD

########################################################

##################### GARCH ################

# norm 1GARCH

spec.garch.norm.11 = ugarchspec(mean.model = list(armaOrder = c(0,0), include.mean = FALSE),