est.fArch: Estimate Functional ARCH Model
In yzhao7322/CurVol: Curve Data Volatility Analysis

Description Usage Arguments Details Value References See Also Examples

est.fArch function estimates the Functional ARCH(q) model by using the Quasi-Maximum Likelihood Estimation method.

1	est.fArch(fdata, basis, q = 1)

`fdata`	The functional data object with N paths.
`basis`	The M-dimensional basis functions.
`q`	The order of the depedence on past squared observations. If it is missing, q=1.

This function estimates the Functional ARCH(q) model:
x_i(t)=σ_i(t)\varepsilon_i(t), for t \in [0,1] and 1≤q i ≤q N,
σ_i^2(t)=ω(t)+ ∑_{j=1}^q \int α_j(t,s) x^2_{i-j}(s)ds.

List of model paramters:

d: d Parameter vector, for intercept function δ.

As: A Matrices, for α operators.

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>.
Cerovecki, C., Francq, C., Hormann, S., Zakoian, J. M. (2019). Functional GARCH models: The quasi-likelihood approach and its applications. Journal of Econometrics. 209(2), 353-375. <doi:10.1016/j.jeconom.2019.01.006>.
Hormann, S., Horvath, L., Reeder, R. (2013). A functional version of the ARCH model. Econometric Theory. 29(2), 267-288. <doi:10.1017/S0266466612000345>.

est.fGarch est.fGarchx diagnostic.fGarch

## Not run: 
# generate discrete evaluations of the FARCH process and smooth them into a functional data object.
yd = dgp.fgarch(grid_point=50, N=200, "arch")
yd = yd$garch_mat
fd = fda::Data2fd(argvals=seq(0,1,len=50),y=yd,fda::create.bspline.basis(nbasis=32))

# extract data-driven basis functions through the truncated FPCA method.
basis_est = basis.est(yd, M=2, "tfpca")$basis

# estimate an FARCH(1) model with basis when M=1.
arch1_est = est.fArch(fd, basis_est[,1])

## End(Not run)