4.1 MGARCH modelling in R
| Command | Description |
|---|---|
DCCtest() |
testing for non-constant correlations |
dccspec() |
creating a DCC specification object prior to fitting |
dccfit() |
estimating (fitting) specified DCC model |
show() |
displays the results of estimated DCC model (mGARCHfit object) |
fitted() |
produces fitted conditional means |
sigma() |
produces fitted conditional standard deviations |
residuals() |
extracts fitted conditional mean residuals |
dccforecast() |
forecasts the future variances and covariances for h-steps ahead |
infocriteria() |
exctrcts information criteria, such as AIC, BIC,… |
coef() |
displays coefficient vector |
likelihood() |
extracts the joint maximum likelihhod |
plot() |
displys \(4\) possible plots by selecting numbers from \(1\) do \(4\) |
nisurface() |
plots news impact surface |
rcor() |
extracts fitted dynamic conditional correlations |
rcov() |
extracts fitted dynamic conditional covariances |
rshape() |
obtains df of multivariate t–distribution |
rskew() |
obtains skewnees of multivariate t–distribution |
Example 12. text here
Solution
Copy the code lines below to the clipboard, paste them into an R Script file, and run them.# Installing and loading new package "rmgarch"
install.packages("rmgarch")
library(rmgarch) # also requires "rugarch" for univariate GARCH
# Downloading the data for Tesla and S&P500 index
getSymbols(c("TSLA","^GSPC"), # tickers for both equities
src="yahoo", # specifying the source of the data
from = as.Date("2021-01-01"), # starting date in the format "yyyy-mm-dd"
to = as.Date("2024-12-31")) # ending date in the format "yyyy-mm-dd"
# Task: Estimate a DCC(1,1) model between Tesla returns and the S&P 500 index
# to study the time-varying beta coefficient.
# Compute Tesla and S&P 500 daily log returns and removing missig values (if any)
tesla_returns <- na.omit(dailyReturn(Cl(TSLA), type = "log"))
sp500_returns <- na.omit(dailyReturn(Cl(GSPC), type = "log"))
# Merge returns into a single dataframe with renamed columns
# Keep only the dates that exist in both returns
returns_data <- merge(tesla_returns, sp500_returns, join = "inner")
colnames(returns_data) <- c("Tesla", "SP500")
# Univariate GARCH(1,1) should be specified in the first step prior do DCC specification
univariate <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1, 1)), # standard GARCH(1,1)
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE), # constant mean
distribution.model = "norm" # normal distribution of innovations
)
# Multivariate DCC-GARCH specification in the secon step
multivariate <- dccspec(
uspec = multispec(replicate(2, univariate)), # same univariate GARCH for both assets
dccOrder = c(1, 1), # DCC(1,1) order
distribution = "mvnorm" # multivariate normal innovations
)
# Fitting the DCC(1,1)-GARCH model
dcc_fit <- dccfit(multivariate, data = as.matrix(returns_data))
print(dcc_fit) # Display estimation results
# Extracting dynamic correlations
dcc_rho <- rcor(dcc_fit) # 1004 correlation matrices 2x2
rho_t <- dcc_rho[1, 2, ] # extracting off-diagonal elements
rho_t <- xts(rho_t, order.by = index(returns_data))
colnames(rho_t) <- c("rho")
rho_t[1:10] # preview first 10 correlations
# Plotting dynamic correlations
ggplot(rho_t, aes(x = Index, y = rho)) +
geom_line(color = "orchid") +
labs(title = "Dynamic correlations between Tesla and S&P500", x = "Days", y = "Correlation") +
theme_minimal()
# Calculating time-varying beta: beta_t = rho_t * (sigma_TSLA / sigma_SP500)
sigma <- sigma(dcc_fit)
index(sigma) <- as.Date(index(sigma))
sigma_tesla <- sigma[, "Tesla"] # Tesla volatility
sigma_sp500 <- sigma[, "SP500"] # S&P500 volatility
beta_t <- rho_t * (sigma_tesla / sigma_sp500)
# Plotting time-varying betas
colnames(beta_t) <- c("beta")
ggplot(beta_t, aes(x = Index, y = beta)) +
geom_line(color = "orchid") +
labs(title = "Time-varying betas of Tesla", x = "Days", y = "Beta cofficient") +
theme_minimal()
# Finding the dates when beta was less or equal to 1
beta_t[beta_t <= 1]
# Summarizing beta coefficients
quantile(beta_t) # five-number summary\(~~~\)
Example 13. text here
Solution
Copy the code lines below to the clipboard, paste them into an R Script file, and run them.# Downloading the data for BitCoin and S&P500 index starting from January 2020
getSymbols(c("^GSPC","BTC-USD"), # tickers for equity and cryptocurrency
src="yahoo", # specifying the source of the data
from = as.Date("2020-01-01"), # also covers the pandemic period
to = as.Date("2024-12-31"))
# Compute S&P500 and BitCoin daily log returns and removing missing values (if any)
sp500_returns <- na.omit(dailyReturn(Cl(GSPC), type = "log"))
bitcoin_returns <- na.omit(dailyReturn(Cl(`BTC-USD`), type = "log")) # use backticks `` when there is dash (-) in objects name
# Merge returns into a single dataframe with renamed columns
# Keep only the dates that exist in both returns
returns_data <- merge(sp500_returns, bitcoin_returns, join = "inner")
colnames(returns_data) <- c("SP500","BitCoin")
# Estimating one correlation for entire period
cor(returns_data)
# Univariate GARCH(1,1) should be specified in the first step prior do DCC specification
univariate <- ugarchspec(
variance.model = list(model = "sGARCH", garchOrder = c(1, 1)), # standard GARCH(1,1)
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE), # constant mean
distribution.model = "norm" # normal distribution of innovations
)
# Multivariate DCC-GARCH specification in the second step
multivariate1 <- dccspec(
uspec = multispec(replicate(2, univariate)), # same univariate GARCH for both assets
dccOrder = c(1, 1), # DCC(1,1) order
model="DCC", # standard DCC model
distribution = "mvt" # multivariate normal innovations
)
# Fitting the DCC(1,1)-GARCH model
dcc_fit1 <- dccfit(multivariate1, data = as.matrix(returns_data))
print(dcc_fit1) # Display estimation results
# Extracting dynamic correlations
dcc_rho <- rcor(dcc_fit1) # 1257 correlation matrices 2x2
rho_t <- dcc_rho[1, 2, ] # extracting off-diagonal elements
rho_t <- xts(rho_t, order.by = index(returns_data))
colnames(rho_t) <- c("rho")
rho_t[1:10] # preview first 10 correlations
# Plotting dynamic correlations
ggplot(rho_t, aes(x = Index, y = rho)) +
geom_line(color = "orchid") +
labs(title = "Dynamic correlations between S&P500 and BitCoin", x = "Days", y = "Correlation") +
theme_minimal()
# Alternative to dynamic correlation plot
plot(dcc_fit1,which=4)
# Covariance can be also plotted
plot(dcc_fit1,which=3)
# Finding the dates when correlation was less or equal to 0
rho_t[rho_t <= 0]
sum(rho_t <= 0)/length(rho_t)*100
# Two functions are defined that return two data frames: tidy and glance
# This is required as modelsummary() command does not support MGARCH model
tidy.DCCfit<- function(x, ...) {
s <- x@mfit$matcoef
ret <- data.frame(
term = row.names(s),
estimate = s[, " Estimate"],
std.error = s[, " Std. Error"],
statistic = s[, " t value"],
p.value = s[,"Pr(>|t|)"])
ret
}
glance.DCCfit <- function(x, ...) {
s <- x@mfit
ret <- data.frame(
Likelihood = round(s$llh, digits=2),
Akaike=round(infocriteria(x)[1], digits=4),
Bayes=round(infocriteria(x)[2], digits=4))
ret
}
# Fitting an asymmetric DCC(1,1) model by incorporating univariate GJR-GARCH models
univariate2 <- ugarchspec(
variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)), # asymmetric GARCH(1,1)
mean.model = list(armaOrder = c(0, 0), include.mean = TRUE), # constant mean
distribution.model = "std", # t- distribution of innovations
fixed.pars=list(shape=5))
# Same univariate GARCH for both assets
multivariate2 <- dccspec(uspec = multispec(replicate(2, univariate2)),
dccOrder = c(1, 1), # DCC(1,1) order
model="DCC", # asymmetric DCC model
distribution = "mvt") # multivariate t-distribution
dcc_fit2 <- dccfit(multivariate2, data = as.matrix(returns_data))
modelsummary(list("DCC(1,1) model"=dcc_fit1,"Asymmetric DCC(1,1)"=dcc_fit2),
stars = c('*' = 0.1, '**' = 0.05, '***' = 0.01), # customizing significance levels
fmt=4)\(~~~\)