predict.garch1c1 {cvar}

Prediction for GARCH(1,1) time series

Description

Predict GARCH(1,1) time series.

Usage

## S3 method for class 'garch1c1'
predict(object, n.ahead = 1, Nsim = 1000, eps, sigmasq, seed = NULL, ...)

Arguments

object

an object from class "garch1c1".

n.ahead

maximum horizon (lead time) for prediction.

Nsim

number of Monte Carlo simulations for simulation based quantities.

eps

the time series to predict, only the last value is used.

sigmasq

the (squared) volatilities, only the last value is used.

seed

an integer, seed for the random number generator.

...

currently not used.

Details

Plug-in prediction intervals and predictive distributions are obtained by inserting the predicted volatility in the conditional densities. For predictions more than one lag ahead these are not the real predictive distributions but the prediction intervals are usually adequate.

For simulation prediction intervals we generate a (large) number of continuations of the given time series. Prediction intervals can be based on sample quantiles. The generated samples are stored in the returned object and can be used for further exploration of the predictive distributions. dist_sim$eps contains the simulated future values of the time series and dist_sim$h the corresponding (squared) volatilities. Both are matrices whose i-th rows contain the predicted quantities for horizon i.

The random seed at the start of the simulations is saved in the returned object. A speficific seed can be requested with argument seed. In that case the simulations are done with the specified seed and the old state of the random number generator is restored before the function returns. This setup is similar to sim_garch1c1.

Value

an object from S3 class "predict_garch1c1" containing the following components:

eps

point predictions (conditional expectations) of the time series (equal to zero for pure GARCH).

h

point predictions (conditional expectations)of the squared volatilities.

model

the model.

call

the call.

pi_plugin

Prediction intervals for the time series, based on plug-in distributions, see Details.

pi_sim

Simulation based prediction intervals for the time series, see Details.

dist_sim

simulation samples from the predictive distributions of the time series and the volatilties.

Note

This function is under development and may be changed.

Examples

op <- options(digits = 4)

## set up a model and simulate a time series
mo <- GarchModel(omega = 0.4, alpha = 0.3, beta = 0.5)
a1 <- sim_garch1c1(mo, n = 1000, n.start = 100, seed = 20220305)

## predictions for T+1,...,T+5 (T = time of last value)
## Nsim is small to reduce the load on CRAN, usually Nsim is larger.
a.pred <- predict(mo, n.ahead = 5, Nsim = 1000, eps = a1$eps,
  sigmasq = a1$h, seed = 1234)

## preditions for the time series
a.pred$eps

## PI's for eps - plug-in and simulated
a.pred$pi_plugin
a.pred$pi_sim

## a DIY calculation of PI's using the simulated sample paths
t(apply(a.pred$dist_sim$eps, 1, function(x) quantile(x, c(0.025, 0.975))))

## further investigate the predictive distributions
t(apply(a.pred$dist_sim$eps, 1, function(x) summary(x)))

## compare predictive densities for horizons 2 and 5:
h2 <- a.pred$dist_sim$eps[2, ]
h5 <- a.pred$dist_sim$eps[5, ]
main <- "Predictive densities: horizons 2 (blue) and 5 (black)"
plot(density(h5), main = main)
lines(density(h2), col = "blue")

## predictions of sigma_t^2
a.pred$h

## plug-in predictions of sigma_t
sqrt(a.pred$h)

## simulation predictive densities (PD's) of sigma_t for horizons 2 and 5:
h2 <- sqrt(a.pred$dist_sim$h[2, ])
h5 <- sqrt(a.pred$dist_sim$h[5, ])
main <- "PD's of sigma_t for horizons 2 (blue) and 5 (black)"
plot(density(h2), col = "blue", main = main)
lines(density(h5))

## VaR and ES for different horizons
cbind(h = 1:5,
      VaR = apply(a.pred$dist_sim$eps, 1, function(x) VaR(x, c(0.05))),
      ES = apply(a.pred$dist_sim$eps, 1, function(x) ES(x, c(0.05))) )

## fit a GARCH(1,1) model to exchange rate data and predict
gmo1 <- fGarch::garchFit(formula = ~garch(1, 1), data = fGarch::dem2gbp,
  include.mean = FALSE, cond.dist = "norm", trace = FALSE)
mocoef <- gmo1@fit$par
mofitted <- GarchModel(omega = mocoef["omega"], alpha = mocoef["alpha1"],
  beta = mocoef["beta1"])
gmo1.pred <- predict(mofitted, n.ahead = 5, Nsim = 1000, eps = gmo1@data,
  sigmasq = gmo1@h.t, seed = 1234)
gmo1.pred$pi_plugin
gmo1.pred$pi_sim

op <- options(op) # restore options(digits)