arch.univariate.FIGARCH¶

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

FIGARCH model

Parameters:¶

p: int = 1¶

Order of the symmetric innovation

q: int = 1¶

Order of the lagged (transformed) conditional variance

power: float = 2.0¶

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 = 1000¶

Truncation point to use in ARCH(\(\infty\)) representation. Default is 1000.

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
update(index, parameters, resids, sigma2, ...)	Compute the variance for a single observation
variance_bounds(resids[, power])	Construct loose bounds for conditional variances.

Properties

name	The name of the volatility process
num_params	The number of parameters in the model
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
updateable	Flag indicating that the volatility process supports update
volatility_updater	Get the volatility updater associated with the volatility process