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Innovation distribution for ARMA-GARCH/APARCH models

Source: R/dist.R
gs_dist.Rd

Builds the object that describes the innovation distribution \(z_t\) of the model. The built in families gs_stable(), gs_gev() and gs_gat() are created with this constructor, and a user defined distribution gets exactly the same fitting, simulation, stationarity and forecasting machinery.

Usage

gs_dist(
  name,
  log_density,
  random,
  cdf,
  quantile,
  par_names,
  lower,
  upper,
  start,
  default_delta,
  max_power,
  mean,
  valid = NULL,
  aparch_moment = NULL,
  start_fun = NULL,
  check = TRUE
)

Arguments

name

Character, name of the distribution.

log_density

function(z, par) returning \(\log f(z)\), vectorized in z, and -Inf outside the support.

random

function(n, par) returning n draws.

cdf

function(q, par), distribution function.

quantile

function(p, par), quantile function.

par_names

Character vector with the parameter names. They must not clash with model parameter names (mu, ar1, omega, alpha1, ...).

lower, upper, start

Numeric vectors named by par_names: box bounds used in estimation and the default start value (strictly inside).

default_delta

Power \(\delta\) used when the model is a GARCH (not APARCH) model, e.g. 2 for a variance recursion, 1 for a scale recursion.

max_power

function(par) returning the supremum of the powers \(\delta\) with \(E|z|^\delta < \infty\) (Inf if all moments exist).

mean

function(par) returning \(E[z]\); it must return NA explicitly where the mean does not exist.

valid

Optional function(par) returning TRUE/FALSE for constraints that are not box bounds.

aparch_moment

Optional function(gamma, delta, par) returning \(E(|z| - \gamma z)^\delta\) in closed form. It may return NA where no closed form is available; numerical integration is then used.

start_fun

Optional function(z) returning data driven start values from standardized data.

check

Logical, run gs_check_dist() at start.

Value

An object of class gs_dist.

Details

All functions work on the standardized innovation (location 0, scale 1). z is a numeric vector and par a numeric vector named by par_names.

Every field is required except valid, aparch_moment and start_fun, which have well defined fallbacks. In particular max_power and mean have no defaults: the user must state when moments exist, so that fitting never fails silently. Unless check = FALSE, gs_check_dist() runs a self check at start and stops with an error naming the failing field.

See also

gs_check_dist(), gs_stable(), gs_gev(), gs_gat()

Examples

# Normal innovations written as a user defined distribution
norm <- gs_dist(
  name = "norm",
  log_density = function(z, par) dnorm(z, log = TRUE),
  random = function(n, par) rnorm(n),
  cdf = function(q, par) pnorm(q),
  quantile = function(p, par) qnorm(p),
  par_names = character(0),
  lower = numeric(0), upper = numeric(0), start = numeric(0),
  default_delta = 2,
  max_power = function(par) Inf,
  mean = function(par) 0)
norm
#> Innovation distribution: norm 
#> No parameters
#> GARCH mode delta: 2