| TARCH {tsDyn} | R Documentation |
Treshold AutoRegressive Conditionally Heteroschedastic model
tarch(x, m, d=1, steps=d, series, coef, thDelay=0, control=list(), ...)
x |
time series |
m, d, steps |
embedding dimension, time delay, forecasting steps |
series |
time series name (optional) |
coef |
vector of starting coefficients values. If missing, they are randomly generated from the log-normal distribution |
thDelay |
time delay value for thresholding |
control, ... |
additional parameters to be passed to optim |
Treshold-ARCH model:
x[t] = sigma[t] eps[t]
with eps[t] standard white noise, and sigma[t] conditional standard deviation which takes the form:
sigma2[t+steps] = ( b[0,0] + sum_j b[0,j] sigma2[t-(j-1)d] ) * (Z[t] <= 0) + ( b[1,0] + sum_j b[1,j] sigma2[t-(j-1)d] ) * (Z[t] > 0)
and Z[t] threshold variable defined as
Z[t] = x[t-thDelay*d].
The model is estimated by Conditional Maximum Likelihood, with
positivity of parameters restriction (strict for
b[0,0] and b[1,0]), using the L-BFGS-B
provided by the optim function.
Standard errors provided in the summary are asymptoticals.
No model specific plots are produced by the plot method.
An object of class tarch.
Antonio, Fabio Di Narzo
Threshold Arch Models and asymmetries in volatility, R. Rabemanajara and J. M. Zakoian, Journal of Applied Econometrics, vol. 8 (1993)
Threshold heteroschedastic models, J. M. Zakoian, D. P. INSEE (1991)
#
#Taken from tseries::garch man page
#
n <- 1100
a <- c(0.1, 0.5, 0.2) # ARCH(2) coefficients
e <- rnorm(n)
x <- double(n)
x[1:2] <- rnorm(2, sd = sqrt(a[1]/(1.0-a[2]-a[3])))
for(i in 3:n) # Generate ARCH(2) process
{
x[i] <- e[i]*sqrt(a[1]+a[2]*x[i-1]^2+a[3]*x[i-2]^2)
}
x <- ts(x[101:1100])
x.tarch <- tarch(x, m=2)
summary(x.tarch)