rbtimeseries {ExtremeRisks}

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 dist="studentT" specifies a Student-t family of distributions. See Details.

type

A string specifying the type of time series. By default type="AR" specifies a linear Auto-Regressive time series. See Details.

copula

A string specifying the type copula to be used. By default copula="Gumbel" specifies the Gumbel copula. See Details.

par

A list of p parameters to be specified for the bivariate time series parametric family. See Details.

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" and copula="studentT") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) and dependence parameter dep\in(-1,1). The parameters are specified as par <- list(corr, df, dep);

  • Asymmetric Student-t (dist="AStudentT" and copula="studentT") with marginal parameters (equal for both distributions): \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) and dependence parameter dep\in(-1,1). The paraters are specified as par <- 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" and copula="Gumbel" or copula="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 parameter dep (dep>0 if copula="Gumbel" or dep\in(-1,1) if copula="Gaussian"). The parameters are specified as par <- list(corr, scale, shape, smooth, dep).

  • symmetric Pareto (dist="double-Pareto" and copula="Gumbel" or copula="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 parameter dep (dep>0 if copula="Gumbel" or dep\in(-1,1) if copula="Gaussian"). The parameters are specified as par <- 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 Gaussian dist="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 parameter dep\in(-1,1), Student-t (copula="studentT") with dependence parameters dep\in(-1,1) and \nu>0 (degrees of freedom), Gumbel (copula="Gumbel") with dependence parameter dep>0. The parameters are specified as par <- list(alpha0, alpha1, beta, dep) or par <- 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, [e-mail], http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, [e-mail], http://ensai.fr/en/equipe/stupfler-gilles/

References

Brockwell, Peter J., and Richard A. Davis. (2016). Introduction to time series and forecasting. Springer.

Padoan A.S. and Stupfler, G. (2020). Extreme expectile estimation for heavy-tailed time series. arXiv e-prints arXiv:2004.04078, https://arxiv.org/abs/2004.04078.

See Also

rtimeseries, expectiles

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")