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

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