locpol_spec {fEGarch} | R Documentation |
Specification of Nonparametric Local Polynomial Models
Description
Specify the nonparametric local polynomial model part in a semiparametric volatility model.
Usage
locpol_spec(
poly_order = c(3, 1),
kernel_order = c(1, 0, 2, 3),
boundary_method = c("extend", "shorten"),
bwidth = NULL
)
Arguments
poly_order |
a single numeric value, in detail either |
kernel_order |
a single numeric value representing
the smoothness of the underlying kernel function;
available are |
boundary_method |
a single character value indicating
the smoothing concept to use at boundary points; for
|
bwidth |
the smoothing bandwidth; for NULL, i.e. the default, an automated bandwidth selection is employed; otherwise a single numeric value between 0 and 0.5 must be provided. |
Details
Assume that a time series \{r_t\}
, t=1,\dots,n
, follows
r_t = \mu + \sigma_t \eta_t,
where \mu = E(r_t)
and \eta_t
are independent and identically
distributed random variables with mean zero and variance one. \sigma_t > 0
are total volatilities composed of s(x_t)
, a smooth, deterministic
scale function in the unconditional variance over time (with x_t
being
the rescaled time on the interval [0, 1]
), and of \lambda_t
,
the conditional standard deviation in \zeta_t=\lambda_t\eta_t
, so that
\sigma_t = s(x_t)\lambda_t
, or alternatively r_t = \mu + s(x_t)\zeta_t
.
It is assumed that the unconditional variance of the \zeta_t
is one.
The package's estimation of \sigma_t
is based on the following relations:
r_t^{*} = r_t - \mu
,
y_t=\ln\left[\left(r_t^{*}\right)^2\right]
,
C_{\mu}=E\left[\ln\left(\zeta_t^2\right)\right]
,
m(x_t) = \ln\left[s^2 (x_t)\right] + C_{\mu}
,
\xi_t = \ln\left(\zeta_t^2\right) - C_{\mu}
, so that
y_t = m(x_t)+\xi_t,
where m
describes a smooth, deterministic trend in y_t
.
Nonparametric estimation of m
and subsequent retransformation
allows to obtain a suitable estimate of the scale function s
in r_t
. Following Feng et al. (2022) and Letmathe et al. (2023),
we employ local polynomial regression with automatically selected
bandwidth (specially for the time-series context). The function
locpol_spec
allows to set the basic characteristics of the
local polynomial estimator considered, like the order of polynomial
used in the local regressions, and the kernel function order. After
the scale function has been estimated, a zero-mean GARCH-type model
can be fitted to the estimated \zeta_t
.
Depending on whether \zeta_t
is assumed to follow a short-memory
or a long-memory model, the bandwidth selection algorithm in the
local polynomial regression step differs and follows either
Feng et al. (2022) and Letmathe et al. (2023). The algorithm
selection is done automatically based on the remaining model
specifications in the call to the estimation functions like
fEGarch
.
Value
An object of class "locpol_spec"
is returned.
References
Feng, Y., Gries, T., Letmathe, S., & Schulz, D. (2022). The smoots Package in R for Semiparametric Modeling of Trend Stationary Time Series. The R Journal, 14(1), 182-195. URL: https://journal.r-project.org/articles/RJ-2022-017/.
Letmathe, S., Beran, J., & Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods, 53(22), 7914–7926. DOI: 10.1080/03610926.2023.2276049.
Examples
locpol_spec()
locpol_spec(poly_order = 1)
locpol_spec(kernel_order = 2)