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 x 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), ν>0 (degrees of freedom) specified by `par=c(corr, df)`;

• symmetric Frechet (`dist="double-Frechet"`) with parameters phi in(-1,1) (autoregressive coefficient), σ>0 (scale), α>0 (shape), θ (movingaverage coefficient), specified by `par=c(corr, scale, shape, smooth)`;

• symmetric Pareto (`dist="double-Pareto"`) with parameters phi in(-1,1) (autoregressive coefficient), σ>0 (scale), α>0 (shape), θ (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, α_0, α_1, β 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), σ>0 (scale), α>0 (shape) specified by `par=c(corr, scale, shape)`.

### Value

A vector of (1 x 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]