tsDyn (version 0.5-7)

TARCH: Treshold-ARCH model

Description

Treshold AutoRegressive Conditionally Heteroschedastic model

Usage

tarch(x, m, d=1, steps=d, series, coef, thDelay=0, control=list(), ...)

Arguments

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

Value

• An object of class tarch.

Details

Treshold-ARCH model: $$x_t = \sigma_t \epsilon_t$$ with $\epsilon_t$ standard white noise, and $\sigma_t$ conditional standard deviation which takes the form: $$\sigma^2_{t+s} = [b_{0,0} + \sum_{j=1}^m b_{0,j} \sigma^2_{t-(j-1)d}] I(Z_t \leq 0) + [b_{1,0} + \sum_{j=1}^m b_{1,j} \sigma^2_{t-(j-1)d}] I(Z_t > 0)$$

and $Z_t$ threshold variable defined as $Z_t = x_{t-thD\cdot d}$. 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.

References

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)

See Also

setar, lstar

Examples

#
#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)