Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model

Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is a statistical model used in analyzing time-series data where the variance error is believed to be serially autocorrelated. GARCH models assume that the variance of the error term follows an autoregressive moving average process.

GARCH (p, q) model (where p is the order of the GARCH terms \(\sigma^{2}\) and q is the order of the ARCH terms \(\epsilon^{2}\)) is a model which \(\epsilon_{t}\), the error terms, can be split into a stochastic piece \(z_{t}\) and a time-dependent standard deviation \(\sigma_{t}\) characterizing the typical size of the terms so that \(\epsilon_{t}=\sigma_{t}z_{t}\). The random variable \(z_{t}\) is a strong white noise process while \(\sigma_{t}^{2}\) is an ARMA process, i.e.,

Keep in Mind

• Data should be properly formatted for estimation as a time-series. See creating a time series data set. If not, you may fail to execute or receive erroneous output.
• GARCH is appropriate for time series data where the variance of the error term is serially autocorrelated following an autoregressive moving average process.

Also Consider

• GARCH can be used to help predict the volatility of returns on financial assets.
• GARCH is useful to assess risk and expected returns for assets that exhibit clustered periods of volatility in returns.
• If an autoregressive(AR) model is assumed for the error variance, the model is an autoregressive conditional heteroskedasticity (ARCH) model. For more information on GARCH models, see Wikipedia: ARCH. For information about estimating an ARCH model, see LOST: ARCH models.

Implementations

Julia

The ARCHModels.jl package offers a variety of ARCH-model simulation, estimation, and forecasting tools.

We start by loading the package.

# Load necessary packages 
using ARCHModels

Next, we use the simulate function to specify a GARCH{1,1} model with coefficient parameters a0, b1, and a1, and then simulate a realization of the specified data-generating process with 1000 observations. Here the a0 parameter corresponds to the intercept term, b1 corresponds to the \(p=1\) lag coefficient in GARCH(\(p,q\)), and a1 corresponds to the \(q=1\) lag coefficient.

# Simulate a GARCH(1,1) process
a0 = 0.2
a1 = 0.5
b1 = 0.3
garch11sim = simulate(GARCH{1,1}([a0, b1, a1]), 1000)

Lastly, we use the fit function to fit an GARCH{1,1} model to the generated series contained in the data attribute of the UnivariateARCHModel object we named garch11sim in the above code chunk.

# Fit GARCH(1,1) model to simulated data
fit(GARCH{1,1}, garch11sim.data)

Python

# setup
from random import gauss
from random import seed
from matplotlib import pyplot
from arch import arch_model
import numpy as np
# seed the process
np.random.seed(1)
# Simulating a GARCH(1, 1) process
a0 = 0.2
a1 = 0.5
b1 = 0.3
n = 1000
w = np.random.normal(size=n)
eps = np.zeros_like(w)
sigsq = np.zeros_like(w)
for i in range(1, n):
    sigsq[i] = a0 + a1*(eps[i-1]**2) + b1*sigsq[i-1]
    eps[i] = w[i] * np.sqrt(sigsq[i])
model = arch_model(eps)
model_fit = model.fit()
print(model_fit.summary)

R

# setup
library(fGarch)

# seed pseudorandom number generator
set.seed(1)

# Simulating a GARCH(1,1) process
a0 <- 0.2
a1 <- 0.5
b1 <- 0.3
obs <- 1000
eps <- rep(0, obs)
sigsq <- rep(0,obs)
for (i in 2:obs) {
  sigsq[i] = a0 + a1*(eps[i-1]^2) + b1*sigsq[i-1]
  eps[i] <- rnorm(1)*sqrt(sigsq[i])}

# fit the model
garch.fit <- garchFit(~garch(1,1), data = eps, trace = F)
summary(garch.fit)