TARCH                 package:tsDyn                 R Documentation

_T_r_e_s_h_o_l_d-_A_R_C_H _m_o_d_e_l

_D_e_s_c_r_i_p_t_i_o_n:

     Treshold AutoRegressive Conditionally Heteroschedastic model

_U_s_a_g_e:

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

_A_r_g_u_m_e_n_t_s:

       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'

_D_e_t_a_i_l_s:

     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.

_V_a_l_u_e:

     An object of class 'tarch'.

_A_u_t_h_o_r(_s):

     Antonio, Fabio Di Narzo

_R_e_f_e_r_e_n_c_e_s:

     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)

_S_e_e _A_l_s_o:

     'setar', 'lstar'

_E_x_a_m_p_l_e_s:

     #
     #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)

