rbtimeseries {ExtremeRisks} | R Documentation |
Simulation of Two-Dimensional Temporally Dependent Observations
Description
Simulates samples from parametric families of bivariate time series models.
Usage
rbtimeseries(ndata, dist="studentT", type="AR", copula="Gumbel", par, burnin=1e+03)
Arguments
ndata |
A positive interger specifying the number of observations to simulate. |
dist |
A string specifying the parametric family of the innovations distribution. By default |
type |
A string specifying the type of time series. By default |
copula |
A string specifying the type copula to be used. By default |
par |
A list of |
burnin |
A positive interger specifying the number of initial observations to discard from the simulated sample. |
Details
For a time series class (type
), with a parametric family (dist
) for the innovations, a sample of size ndata
is simulated. See for example Brockwell and Davis (2016).
The available categories of bivariate time series models are: Auto-Regressive (
type="AR"
), Auto-Regressive and Moving-Average (type="ARMA"
), Generalized-Autoregressive-Conditional-Heteroskedasticity (type="GARCH"
) and Auto-Regressive.With AR(1) times series the available families of distributions for the innovations and the dependence structure (copula) are:
Student-t (
dist="studentT"
andcopula="studentT"
) with marginal parameters (equal for both distributions):\phi\in(-1,1)
(autoregressive coefficient),\nu>0
(degrees of freedom) and dependence parameterdep\in(-1,1)
. The parameters are specified aspar <- list(corr, df, dep)
;Asymmetric Student-t (
dist="AStudentT"
andcopula="studentT"
) with marginal parameters (equal for both distributions):\phi\in(-1,1)
(autoregressive coefficient),\nu>0
(degrees of freedom) and dependence parameterdep\in(-1,1)
. The paraters are specified aspar <- list(corr, df, dep)
. Note that in this case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details;
With ARMA(1,1) times series the available families of distributions for the innovations and the dependence structure (copula) are:
symmetric Pareto (
dist="double-Pareto"
andcopula="Gumbel"
orcopula="Gaussian"
) with marginal parameters (equal for both distributions):\phi\in(-1,1)
(autoregressive coefficient),\sigma>0
(scale),\alpha>0
(shape),\theta
(movingaverage coefficient), and dependence parameterdep
(dep>0
ifcopula="Gumbel"
ordep\in(-1,1)
ifcopula="Gaussian"
). The parameters are specified aspar <- list(corr, scale, shape, smooth, dep)
.symmetric Pareto (
dist="double-Pareto"
andcopula="Gumbel"
orcopula="Gaussian"
) with marginal parameters (equal for both distributions):\phi\in(-1,1)
(autoregressive coefficient),\sigma>0
(scale),\alpha>0
(shape),\theta
(movingaverage coefficient), and dependence parameterdep
(dep>0
ifcopula="Gumbel"
ordep\in(-1,1)
ifcopula="Gaussian"
). The parameters are specified aspar <- list(corr, scale, shape, smooth, dep)
. Note that in this case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details;
With ARCH(1)/GARCH(1,1) time series the distribution of the innovations are symmetric Gaussian (
dist="Gaussian"
) or asymmetric Gaussiandist="AGaussian"
. In both cases the marginal parameters (equal for both distributions) are:\alpha_0
,\alpha_1
,\beta
. In the asymmetric Gaussian case the tail index of the lower and upper tail of the first marginal are different, see Padoan and Stupfler (2020) for details. The available copulas are: Gaussian (copula="Gaussian"
) with dependence parameterdep\in(-1,1)
, Student-t (copula="studentT"
) with dependence parametersdep\in(-1,1)
and\nu>0
(degrees of freedom), Gumbel (copula="Gumbel"
) with dependence parameterdep>0
. The parameters are specified aspar <- list(alpha0, alpha1, beta, dep)
orpar <- list(alpha0, alpha1, beta, dep, df)
.
Value
A vector of (2 \times n)
observations simulated from a specified bivariate time series model.
Author(s)
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@univ-angers.fr, https://math.univ-angers.fr/~stupfler/
References
Brockwell, Peter J., and Richard A. Davis. (2016). Introduction to time series and forecasting. Springer.
Anthony C. Davison, Simone A. Padoan and Gilles Stupfler (2023). Tail Risk Inference via Expectiles in Heavy-Tailed Time Series, Journal of Business & Economic Statistics, 41(3) 876-889.
See Also
Examples
# Data simulation from a 2-dimensional AR(1) with bivariate Student-t distributed
# innovations, with one marginal distribution whose lower and upper tail indices
# that are different
tsDist <- "AStudentT"
tsType <- "AR"
tsCopula <- "studentT"
# parameter setting
corr <- 0.8
dep <- 0.8
df <- 3
par <- list(corr=corr, dep=dep, df=df)
# sample size
ndata <- 2500
# Simulates a sample from an AR(1) model with Student-t innovations
data <- rbtimeseries(ndata, tsDist, tsType, tsCopula, par)
# Extreme expectile estimation
plot(data, pch=21)
plot(data[,1], type="l")
plot(data[,2], type="l")