CurVol: Curve Data Volatility Analysis

dgp.fgarch

R Documentation

Data Generating Process - Functional ARCH/GARCH Process

Description

dgp.fgarch function generates functional curve data following the functional ARCH(1) or GARCH(1,1) process.

Usage

dgp.fgarch(grid_point, N, type, alpha_par = NULL, beta_par = NULL)

Arguments

`grid_point`	The number of grid point in each curve observation.
`N`	The sample size.
`type`	A string to switch data generating process between "arch" - functional ARCH and "garch" - functional GARCH.
`alpha_par`	The ARCH kernel coefficient function in the conditional volatility equation. If it is missing, "`12 * t * (1-t) * s * (1-s)`" is used to generate FGARCH, and "`16 * t * (1-t) * s * (1-s)`" is used to generate FARCH, for `t\in[0,1]` and `s\in[0,1]`.
`beta_par`	The GARCH kernel coefficient function in the conditional volatility equation. If it is missing, "`12 * t * (1-t) * s * (1-s)`" is used to generate FGARCH, for `t\in[0,1]` and `s\in[0,1]`.

Details

If x_i(t) follows an FARCH(1) process,
x_i(t)=\sigma_i(t)\varepsilon_i(t), t \in [0,1],
\sigma_i^2(t)=\omega(t)+\int \alpha(t,s) x^2_{i-1}(s)ds.

If x_i(t) follows an FGARCH(1,1) process,
x_i(t)=\sigma_i(t)\varepsilon_i(t), t \in [0,1],
\sigma_i^2(t)=\omega(t)+\int \alpha(t,s) x^2_{i-1}(s)ds+\int \beta(t,s) \sigma^2_{i-1}(s)ds,
where the innovation \varepsilon_i(t) follows an Ornstein–Uhlenbeck process dgp.fiid, and the constant coefficient \omega(t)=0.1t(1-t).

Value

List of generated processes:

garch_mat: FARCH/GARCH sequences, where the finite realization of curves are stored in columns;

sigma_mat: Conditional volatility sequences, where the finite realization of curves are stored in columns.

References

Hormann, S., Horvath, L., Reeder, R. (2013). A functional version of the ARCH model. Econometric Theory. 29(2), 267-288. <doi:10.1017/S0266466612000345>.
Aue, A., Horvath, L., F. Pellatt, D. (2017). Functional generalized autoregressive conditional heteroskedasticity. Journal of Time Series Analysis. 38(1), 3-21. <doi:10.1111/jtsa.12192>.

Examples

# generate discrete evaluations of 100 FARCH curves that each curve is realized on 50 grid points.
yd = dgp.fgarch(50, 100, "garch")
yd_garch = yd$garch_mat

# smooth discrete data into functional curves.
fd = fda::Data2fd(argvals=seq(0,1,len=50),y=yd_garch,fda::create.bspline.basis(nbasis=32))

yzhao7322/CurVol documentation built on June 11, 2025, 8:30 p.m.