Fit an ARMA-GARCH/APARCH model by maximum likelihood

Estimates the model of gs_spec() by conditional maximum likelihood. The data is rescaled internally to unit standard deviation (as in fGarch) and the estimates are transformed back. Parameters move in plain intervals; stationarity is not imposed but reported after fitting (see gs_stationarity()), with a warning when the persistence is 1 or more.

Usage

gs_fit(
  data,
  spec,
  algorithm = c("sqp", "nlminb"),
  start = NULL,
  hessian = TRUE,
  restarts = 0L,
  control = list()
)

Arguments

data

Numeric vector, the time series (for example log returns).

spec

A gs_spec object.

algorithm

"sqp" (sequential quadratic programming via Rsolnp::solnp(), Galanos and Ye 2025) or "nlminb".

start

Optional named vector of start values on the original scale, for example coef(previous_fit) in rolling windows. Missing names are filled with the default start values.

hessian

Logical, compute standard errors. Set FALSE to save time when only point estimates are needed.

restarts

Integer, number of additional optimizer runs from perturbed start values; the fit with the highest log likelihood is kept. A few restarts (for example 3 to 5) make the estimate robust to the local optima of multimodal surfaces, such as ARMA-GARCH models with heavy-tailed innovations, at a proportional increase in time. The default 0 keeps the single run from the data-driven start.

control

List passed to the optimizer.

Value

An object of class gs_fit with methods coef(), vcov(), logLik(), residuals(), sigma(), fitted(), predict() and print().

Details

Standard errors come from a numerical Hessian (central differences with steps adapted to the curvature). When a parameter sits on a bound, or a differentiation step leaves the parameter space, the affected standard errors are NA and a warning explains why.

These Hessian standard errors are asymptotic and rely on the usual regularity conditions. They become unreliable near a boundary of the parameter space, even when not exactly on it: the stable tail index approaching 2 (where the skewness is no longer identified), the GEV shape approaching \(\pm 0.5\), or an APARCH asymmetry approaching \(\pm 1\). The GEV support depends on its shape parameter, so for GEV innovations the usual maximum likelihood confidence intervals are not guaranteed valid; prefer the parametric bootstrap of gs_bootstrap() there and in the near-boundary cases above. The fit records convergence diagnostics in $diagnostics (scaled gradient, Hessian reciprocal condition number and definiteness, optimizer iterations), which flag these situations.

References

Galanos, A. and Ye, Y. (2025). Rsolnp: General Non-Linear Optimization. R package version 2.0.1. https://CRAN.R-project.org/package=Rsolnp

Examples

# \donttest{
if (requireNamespace("fGarch", quietly = TRUE)) {
  data("dem2gbp", package = "fGarch")
  spec <- gs_spec(arma = c(1, 0), garch = c(1, 1), dist = gs_stable())
  fit <- gs_fit(dem2gbp[, 1], spec)
  fit
}
#> ARMA(1,0)-GARCH(1,1) with stable innovations
#>                Estimate Std. Error t value  Pr(>|t|)    
#> mu            0.0141412  0.0074983  1.8859 0.0593073 .  
#> ar1           0.0214632  0.0217327  0.9876 0.3233502    
#> omega         0.0038210  0.0016143  2.3670 0.0179324 *  
#> alpha1        0.0770510  0.0118725  6.4899 8.591e-11 ***
#> beta1         0.8911481  0.0179740 49.5797 < 2.2e-16 ***
#> stable_alpha  1.7462593  0.0331827 52.6256 < 2.2e-16 ***
#> stable_beta  -0.3748142  0.0989306 -3.7887 0.0001515 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Log likelihood: -987.6780   AIC: 1989.3560   BIC: 2028.4707   n = 1974
#> Persistence: 0.9930
#> Convergence: scaled gradient 3.5e-05, Hessian rcond 6.7e-05
# }