# Simulation, estimation and forecasting of Beta-Skew-t-EGARCH models

### Description

This package provides facilities for the simulation, estimation and forecasting of first order Beta-Skew-t-EGARCH models with leverage (one-component and two-component versions), see Harvey and Sucarrat (2014), and Sucarrat (2013).

Let y[t] denote a financial return at time t equal to

y[t] = sigma[t]*epsilon[t]

where sigma[t] > 0 is the scale or volatility (generally not equal to the conditional standard deviation), and where epsilon[t] is IID and t-distributed (possibly skewed) with df degrees of freedom. Then the first order log-volatility specifiction of the one-component Beta-Skew-t-EGARCH model can be parametrised as

sigma[t] = exp(lambda[t]),

lambda[t] = omega + lambdadagger,

lambdadagger[t] = phi1*lambdadagger[t-1] + kappa1*u[t-1] + kappastar*sign[-y]*(u[t-1]+1).

So the scale or volatility is given by sigma[t] = exp(lambda[t]). The omega is the unconditional or long-term log-volatility, phi1 is the GARCH parameter (|phi1| < 1 implies stability), kappa1 is the ARCH parameter, kappastar is the leverage or volatility-asymmetry parameter and u[t] is the conditional score or first derivative of the log-likelihood with respect to lambda. The score u[t] is zero-mean and IID, and (u[t]+1)/(df+1) is Beta distributed when there is no skew in the conditional density of epsilon[t]. The two-component specification is given by

sigma[t] = exp(lambda[t]),

lambda[t] = omega + lambda1dagger + lambda2dagger,

lambda1dagger[t] = phi1*lambdadagger[t-1] + kappa1*u[t-1],

lambda2dagger[t] = phi2*lambdadagger[t-1] + kappa2*u[t-1] + kappastar*sign[-y]*(u[t-1]+1).

The first component, lambda1dagger, is interpreted as the long-term component, whereas the second component, lambda2dagger, is interpreted as the short-term component.

### Details

Package: | betategarch |

Type: | Package |

Version: | 3.3 |

Date: | 2016-10-16 |

License: | GPL-2 |

LazyLoad: | yes |

The two main functions of the package are `tegarchSim`

and `tegarch`

. The first simulates a Beta-Skew-t-EGARCH models whereas the second estimates one. The second object returns an object (a list) of class 'tegarch', and a collection of methods can be applied to this class: `coef.tegarch`

, `fitted.tegarch`

, `logLik.tegarch`

, `predict.tegarch`

, `print.tegarch`

, `residuals.tegarch`

, `summary.tegarch`

and `vcov.tegarch`

. In addition, the output produced by the `tegarchSim`

function and the `fitted.tegarch`

and `residuals.tegarch`

methods are of the Z's ordered observations (`zoo`

) class, which means a range of time-series methods are available for these objects.

### Author(s)

Genaro Sucarrat, http://www.sucarrat.net/

### References

Fernandez and Steel (1998), 'On Bayesian Modeling of Fat Tails and Skewness', Journal of the American Statistical Association 93, pp. 359-371.

Harvey and Sucarrat (2014), 'EGARCH models with fat tails, skewness and leverage'. Computational Statistics and Data Analysis 76, pp. 320-338.

Sucarrat (2013), 'betategarch: Simulation, Estimation and Forecasting of First-Order Beta-Skew-t-EGARCH models'. The R Journal (Volume 5/2), pp. 137-147.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
#simulate 500 observations from model with default parameter values:
set.seed(123)
y <- tegarchSim(500)
#estimate and store as 'mymod':
mymod <- tegarch(y)
#print estimates and standard errors:
print(mymod)
#graph of fitted volatility (conditional standard deviation):
plot(fitted(mymod))
#plot forecasts of volatility 1-step ahead up to 10-steps ahead:
plot(predict(mymod, n.ahead=10))
``` |