Here is an example of implementation using the rugarch package and with to some fake data. The function ugarchfit allows for the inclusion of external regressors in the mean equation (note the use of external.regressors in fit.spec in the code below).
To fix notations, the model is
\begin{align*}
y_t &= \lambda_0 + \lambda_1 x_{t,1} + \lambda_2 x_{t,2} + \epsilon_t, \\
\epsilon_t &= \sigma_t Z_t , \\
\sigma_t^2 &= \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 ,
\end{align*}
where $x_{t,1}$ and $x_{t,2}$ denote the covariate at time $t$,
and
with the "usual" assumptions/requirements on parameters and the innovation process $Z_t$.
The parameter values used in the example are as follows.
## Model parameters
nb.period <- 1000
omega <- 0.00001
alpha <- 0.12
beta <- 0.87
lambda <- c(0.001, 0.4, 0.2)
The image below shows the series of covariate $x_{t,1}$ and $x_{t,2}$ as well as the series $y_t$. The R code used to generate them is provided below.
## Dependencies
library(rugarch)
## Generate some covariates
set.seed(234)
ext.reg.1 <- 0.01 * (sin(2*pi*(1:nb.period)/nb.period))/2 + rnorm(nb.period, 0, 0.0001)
ext.reg.2 <- 0.05 * (sin(6*pi*(1:nb.period)/nb.period))/2 + rnorm(nb.period, 0, 0.001)
ext.reg <- cbind(ext.reg.1, ext.reg.2)
## Generate some GARCH innovations
sim.spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
distribution.model = "norm",
fixed.pars = list(omega = omega, alpha1 = alpha, beta1 = beta))
path.sgarch <- ugarchpath(sim.spec, n.sim = nb.period, n.start = 1)
epsilon <- as.vector(fitted(path.sgarch))
## Create the time series
y <- lambda[1] + lambda[2] * ext.reg[, 1] + lambda[3] * ext.reg[, 2] + epsilon
## Data visualization
par(mfrow = c(3,1))
plot(ext.reg[, 1], type = "l", xlab = "Time", ylab = "Covariate 1")
plot(ext.reg[, 2], type = "l", xlab = "Time", ylab = "Covariate 2")
plot(y, type = "h", xlab = "Time")
par(mfrow = c(1,1))
A fit is done with ugarchfit as follows.
## Fit
fit.spec <- ugarchspec(variance.model = list(model = "sGARCH",
garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(0, 0),
include.mean = TRUE,
external.regressors = ext.reg),
distribution.model = "norm")
fit <- ugarchfit(data = y, spec = fit.spec)
Parameter estimates are
## Results review
fit.val <- coef(fit)
fit.sd <- diag(vcov(fit))
true.val <- c(lambda, omega, alpha, beta)
fit.conf.lb <- fit.val + qnorm(0.025) * fit.sd
fit.conf.ub <- fit.val + qnorm(0.975) * fit.sd
> print(fit.val)
# mu mxreg1 mxreg2 omega alpha1 beta1
#1.724885e-03 3.942020e-01 7.342743e-02 1.451739e-05 1.022208e-01 8.769060e-01
> print(fit.sd)
#[1] 4.635344e-07 3.255819e-02 1.504019e-03 1.195897e-10 8.312088e-04 3.375684e-04
And corresponding true values are
> print(true.val)
#[1] 0.00100 0.40000 0.20000 0.00001 0.12000 0.87000
The following figure shows a parameter estimates with 95% confidence intervals, and the true values. The R code used to generate it is provided is below.
plot(c(lambda, omega, alpha, beta), pch = 1, col = "red",
ylim = range(c(fit.conf.lb, fit.conf.ub, true.val)),
xlab = "", ylab = "", axes = FALSE)
box(); axis(1, at = 1:length(fit.val), labels = names(fit.val)); axis(2)
points(coef(fit), col = "blue", pch = 4)
for (i in 1:length(fit.val)) {
lines(c(i,i), c(fit.conf.lb[i], fit.conf.ub[i]))
}
legend( "topleft", legend = c("true value", "estimate", "confidence interval"),
col = c("red", "blue", 1), pch = c(1, 4, NA), lty = c(NA, NA, 1), inset = 0.01)
