arch.univariate.FIGARCH

class arch.univariate.FIGARCH(p=1, q=1, power=2.0, truncation=1000)[source]

FIGARCH model

Parameters
  • p ({0, 1}) – Order of the symmetric innovation

  • q ({0, 1}) – Order of the lagged (transformed) conditional variance

  • power (float, optional) – Power to use with the innovations, abs(e) ** power. Default is 2.0, which produces FIGARCH and related models. Using 1.0 produces FIAVARCH and related models. Other powers can be specified, although these should be strictly positive, and usually larger than 0.25.

  • truncation (int, optional) – Truncation point to use in ARCH(\(\infty\)) representation. Default is 1000.

num_params

The number of parameters in the model

Type

int

Examples

>>> from arch.univariate import FIGARCH

Standard FIGARCH

>>> figarch = FIGARCH()

FIARCH

>>> fiarch = FIGARCH(p=0)

FIAVGARCH process

>>> fiavarch = FIGARCH(power=1.0)

Notes

In this class of processes, the variance dynamics are

\[h_t = \omega + [1-\beta L - \phi L (1-L)^d] \epsilon_t^2 + \beta h_{t-1}\]

where L is the lag operator and d is the fractional differencing parameter. The model is estimated using the ARCH(\(\infty\)) representation,

\[h_t = (1-\beta)^{-1} \omega + \sum_{i=1}^\infty \lambda_i \epsilon_{t-i}^2\]

The weights are constructed using

\[\begin{split}\delta_1 = d \\ \lambda_1 = d - \beta + \phi\end{split}\]

and the recursive equations

\[\begin{split}\delta_j = \frac{j - 1 - d}{j} \delta_{j-1} \\ \lambda_j = \beta \lambda_{j-1} + \delta_j - \phi \delta_{j-1}.\end{split}\]

When power is not 2, the ARCH(\(\infty\)) representation is still used where \(\epsilon_t^2\) is replaced by \(|\epsilon_t|^p\) and p is the power.

Methods

backcast(resids)

Construct values for backcasting to start the recursion

backcast_transform(backcast)

Transformation to apply to user-provided backcast values

bounds(resids)

Returns bounds for parameters

compute_variance(parameters, resids, sigma2, …)

Compute the variance for the ARCH model

constraints()

Construct parameter constraints arrays for parameter estimation

forecast(parameters, resids, backcast, …)

Forecast volatility from the model

parameter_names()

Names of model parameters

simulate(parameters, nobs, rng[, burn, …])

Simulate data from the model

starting_values(resids)

Returns starting values for the ARCH model

variance_bounds(resids[, power])

Construct loose bounds for conditional variances.

Properties

name

The name of the volatilty process

start

Index to use to start variance subarray selection

stop

Index to use to stop variance subarray selection

truncation

Truncation lag for the ARCH-infinity approximation