rtimeseries {ExtremeRisks} R Documentation

## Simulation of One-Dimensional Temporally Dependent Observations

### Description

Simulates samples from parametric families of time series models.

### Usage

rtimeseries(ndata, dist="studentT", type="AR", 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. par A vector of (1 \times p) parameters to be specified for the univariate 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 time series models are: Auto-Regressive (type="AR"), Auto-Regressive and Moving-Average (type="ARMA"), Generalized-Autoregressive-Conditional-Heteroskedasticity (type="GARCH") and Auto-Regressive and Moving-Maxima (type="ARMAX").

• With AR(1) and ARMA(1,1) times series the available families of distributions for the innovations are:

• Student-t (dist="studentT") with parameters: \phi\in(-1,1) (autoregressive coefficient), \nu>0 (degrees of freedom) specified by par=c(corr, df);

• symmetric Frechet (dist="double-Frechet") with parameters \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), specified by par=c(corr, scale, shape, smooth);

• symmetric Pareto (dist="double-Pareto") with parameters \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape), \theta (movingaverage coefficient), specified by par=c(corr, scale, shape, smooth).

With ARCH(1)/GARCH(1,1) time series the Gaussian family of distributions is available for the innovations (dist="Gaussian") with parameters, \alpha_0, \alpha_1, \beta specified by par=c(alpha0, alpha1, beta). Finally, with ARMAX(1) times series the Frechet families of distributions is available for the innovations (dist="Frechet") with parameters, \phi\in(-1,1) (autoregressive coefficient), \sigma>0 (scale), \alpha>0 (shape) specified by par=c(corr, scale, shape).

### Value

A vector of (1 \times n) observations simulated from a specified time series model.

### 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.

### Examples

# Data simulation from a 1-dimensional AR(1) with univariate Student-t
# distributed innovations

tsDist <- "studentT"
tsType <- "AR"

# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)

# sample size
ndata <- 2500

# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)

# Graphic representation
plot(data, type="l")
acf(data)



[Package ExtremeRisks version 0.0.4 Index]