Simulates samples from parametric families of bivariate time series models.
rbtimeseries(ndata, dist="studentT", type="AR", copula="Gumbel", par, burnin=1e+03)A vector of \((2 \times n)\) observations simulated from a specified bivariate time series model.
A positive interger specifying the number of observations to simulate.
A string specifying the parametric family of the innovations distribution. By default dist="studentT" specifies a
Student-t family of distributions. See Details.
A string specifying the type of time series. By default type="AR" specifies a linear Auto-Regressive time series. See Details.
A string specifying the type copula to be used. By default copula="Gumbel" specifies the Gumbel copula. See Details.
A list of \(p\) parameters to be specified for the bivariate time series parametric family. See Details.
A positive interger specifying the number of initial observations to discard from the simulated sample.
Simone Padoan, simone.padoan@unibocconi.it, http://mypage.unibocconi.it/simonepadoan/; Gilles Stupfler, gilles.stupfler@ensai.fr, http://ensai.fr/en/equipe/stupfler-gilles/
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).
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.
rtimeseries, expectiles
# 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")
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