cultivars.univariate.conditional_variance

Conditional-variance models: GARCH, GJR, EGARCH, and FIGARCH.

Every model here writes the series as a conditional mean plus a shock whose variance moves,

\[y_t = \mu_t + \varepsilon_t, \qquad \varepsilon_t = \sigma_t z_t, \qquad z_t \sim N(0, 1),\]

with \(\mu_t\) a constant, zero, or an ARMA recursion, and \(\sigma_t^2\) one of four recursions on the past shocks:

\[ \begin{align}\begin{aligned}\text{GARCH:}\quad \sigma_t^2 = \omega + \sum_i \alpha_i \varepsilon_{t-i}^2 + \sum_j \beta_j \sigma_{t-j}^2,\\\text{GJR:}\quad \sigma_t^2 = \omega + \sum_i \alpha_i \varepsilon_{t-i}^2 + \sum_k \gamma_k \varepsilon_{t-k}^2\,\mathbb 1\{\varepsilon_{t-k} < 0\} + \sum_j \beta_j \sigma_{t-j}^2,\\\text{EGARCH:}\quad \log\sigma_t^2 = \omega + \sum_i \alpha_i\bigl(|z_{t-i}| - \sqrt{2/\pi}\bigr) + \sum_k \gamma_k z_{t-k} + \sum_j \beta_j \log\sigma_{t-j}^2,\\\text{FIGARCH:}\quad \sigma_t^2 = \frac{\omega}{1 - \beta} + \Bigl[1 - \frac{(1 - \phi L)(1 - L)^d}{1 - \beta L}\Bigr]\varepsilon_t^2.\end{aligned}\end{align} \]

All four estimate the mean and the variance jointly by maximizing

\[\log L = -\tfrac12 \sum_t \bigl(\log 2\pi + \log\sigma_t^2 + \varepsilon_t^2 / \sigma_t^2\bigr),\]

so the reported likelihood is the true joint one rather than the two-step approximation you get from fitting a mean model and then a variance model to its residuals. The mean uses conditional sum of squares rather than the exact filter because the exact filter assumes a constant innovation variance, which is the one thing these models deny.

Two commitments shape the surface. First, the group splits in two, and the split is not cosmetic. GARCH, GJR and EGARCH are finite-order recursions whose shocks decay geometrically, so persistence is a single number, a half-life exists, and – for the level families – so does an unconditional variance. FIGARCH applies a fractional filter whose ARCH(\(\infty\)) weights decay as \(k^{d-1}\) and sum to one: no geometric rate describes them, no half-life is defined, and the process is not covariance stationary for any \(d > 0\). Sharing one persistence implementation across both would produce a plausible number for FIGARCH that means nothing, which is why the base result declares it and GARCHResult and FIGARCHResult answer it separately. Second, a number that would mislead is withheld rather than approximated. The log-variance family has no closed-form unconditional variance in the reported parameters – its stationary level is the mean of a lognormal – so GARCHResult.unconditional_variance returns None for EGARCH rather than omega / (1 - beta); FIGARCH refuses to simulate because no finite burn-in makes a long-memory sample forget its start; and stationarity on the variance side is persistence < 1 alone, so the constant-mean results carry no mean-stationarity mixin, while the ARMA-mean results carry it and keep the two verdicts apart.

Layout. The constant-mean fronts GARCH, GJR, EGARCH and FIGARCH and the ARMA-mean fronts ARGARCH, ARMAGARCH, ARMAGJR, ARMAEGARCH and ARMAFIGARCH are thin: each fixes a family and a mean shape and delegates to _ShortMemoryVarianceModel or _FractionalVarianceModel in _internals, whose __init__ validates the orders through validate_order and the family and mean through validate_choice against Vol. The mean is a _MeanLayer – _LinearMean for a lag regression, _ARMAMean when a moving-average block forces the constrained recursion – and the variance a _VarianceObjective whose is_admissible rejects draws with persistence at or above 0.999 (short memory) or a negative retained weight (long memory) before _linear_variance_recursion, _log_variance_recursion or _arch_infinity_variance in _core runs the path from an ewma_mean_square backcast and _gaussian_negloglik scores it. _maximize_likelihood searches the unconstrained space, omega in logs and the coefficients through softplus or sigmoid. The packed _ShortMemoryVarianceFit or _FractionalVarianceFit is assembled by each result’s _from_fit, and simulate on the short-memory results runs _simulate_conditional_variance. Latent rather than observation-driven volatility is stochastic_volatility; long memory in the level rather than the variance is fractional_integration.

References

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.

Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1779-1801.

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347-370.

Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1), 3-30.

Example

Four families on one series with a leverage effect, ranked by BIC:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> n = 2500
>>> h, e = np.ones(n), np.zeros(n)
>>> for t in range(1, n):
...     h[t] = 0.02 + 0.05 * e[t - 1] ** 2 + 0.85 * h[t - 1]
...     h[t] += 0.10 * e[t - 1] ** 2 * (e[t - 1] < 0)
...     e[t] = np.sqrt(h[t]) * rng.standard_normal()
>>> fits = [GARCH(e).fit(), GJR(e).fit(), EGARCH(e).fit(), FIGARCH(e).fit()]
>>> table = fits[0].compare(*fits[1:], criterion="bic")
>>> table.rows[0][0] in {"GJR(1, 1, 1)", "EGARCH(1, 1, 1)"}, table.rows[-1][0]
(True, 'FIGARCH(1, d, 1)')
>>> [round(fit.persistence, 2) < 1.0 for fit in fits]
[True, True, True, False]
>>> fits[2].unconditional_variance, fits[3].half_life
(None, inf)

Classes

ARGARCH	Autoregressive mean with a symmetric GARCH variance, estimated jointly.
ARMAEGARCH	ARMA mean with a log-variance EGARCH process, estimated jointly.
ARMAFIGARCH	ARMA mean with a fractionally integrated variance, estimated jointly.
ARMAGARCH	ARMA mean with a symmetric GARCH variance, estimated jointly.
ARMAGJR	ARMA mean with a sign-asymmetric GJR variance, estimated jointly.
EGARCH	EGARCH(p, o, q): the recursion runs in logs, so no positivity constraint binds.
FIGARCH	FIGARCH(1, d, 1): long-memory volatility through a fractional filter.
GARCH	Symmetric GARCH(p, q) with an optional AR mean.
GJR	GJR-GARCH(p, o, q): a level model with a sign-asymmetric ARCH term.
ARMAFIGARCHResult	A fractionally integrated variance model under an ARMA conditional mean.
ARMAGARCHResult	A finite-order variance model under an ARMA conditional mean.
FIGARCHResult	A fitted fractionally integrated conditional-variance model.
GARCHResult	A fitted finite-order conditional-variance model.