Forecast an ARMA-GARCH/APARCH model

Forecasts from a fitted model (predict.gs_fit) or from a model with given parameters and a data history (predict.gs_model, useful in rolling windows with fixed parameters).

Usage

# S3 method for class 'gs_fit'
predict(
  object,
  n_ahead = 1,
  level = c(0.01, 0.05),
  n_sim = 10000,
  seed = NULL,
  ...
)

# S3 method for class 'gs_model'
predict(
  object,
  data,
  n_ahead = 1,
  level = c(0.01, 0.05),
  n_sim = 10000,
  seed = NULL,
  ...
)

Arguments

object

A gs_fit or gs_model object.

n_ahead

Forecast horizon.

level

Probabilities of the reported quantiles.

n_sim

Number of simulated paths used when n_ahead > 1.

seed

Optional seed for the simulated paths.

...

Unused.

data

For gs_model, the observed history (numeric vector).

Value

A data frame with columns horizon, location (one step only), mean, mean_method, sigma and one column per level named q_<level>.

Details

One step ahead everything is exact: the conditional location \(m_{T+1} = \mu + \sum a_i x_{T+1-i} + \sum b_j e_{T+1-j}\), the conditional scale \(\sigma_{T+1}\), and the quantiles \(m_{T+1} + \sigma_{T+1} F^{-1}(p)\) (the value at risk at level p is minus the p quantile). For horizons \(k \ge 2\):

• sigma is \((E\,\sigma^\delta_{T+k})^{1/\delta}\) from the APARCH recursion with \(\kappa = E(|z| - \gamma z)^\delta\) (as in fGarch);

• quantiles come from n_sim simulated paths, since the predictive distribution is a scale mixture and has no closed form;

• mean is exact when \(E[z] = 0\) or \(\delta = 1\), otherwise it is the average of the simulated paths (column mean_method).

With stable innovations there is no forecast variance; sigma is a scale. mean is NA when the innovations have no mean.

Examples

spec <- gs_spec(arma = c(1, 0), garch = c(1, 1), dist = gs_stable())
model <- gs_model(spec, mu = 0, ar = 0.1, omega = 0.05, alpha = 0.1, beta = 0.8,
                  dist_par = c(stable_alpha = 1.8, stable_beta = 0))
x <- gs_sim(model, n = 500, seed = 1)$y
predict(model, data = x, n_ahead = 5, n_sim = 2000, seed = 1)
#>   horizon    location          mean mean_method     sigma    q_0.01    q_0.05
#> 1       1 -0.09342497 -9.342497e-02       exact 0.6321559 -2.797025 -1.676901
#> 2       2          NA -9.342497e-03       exact 0.6359273 -2.748175 -1.621558
#> 3       3          NA -9.342497e-04       exact 0.6394230 -3.059095 -1.624391
#> 4       4          NA -9.342497e-05       exact 0.6426629 -2.747105 -1.683188
#> 5       5          NA -9.342497e-06       exact 0.6456660 -3.091593 -1.647842