Use the skew partial moment in analytic GJR forecasts

[shaneraphel]

Summary

Multi-step analytic GJR forecasts filled the future asymmetric shock with sqrt(0.5 * sigma^2). That is E[z^2 1(z<0)] = 1/2, which is right for a symmetric distribution and wrong for a skewed one. SkewStudent.partial_moment(2, 0) is about 0.55 for a typical equity skew, so the persistence should be alpha + k * gamma + beta, not alpha + 0.5 * gamma + beta.

MIDASHyperbolic with asymmetry wrote the same 0.5 into future indicator rows. Simulation and bootstrap forecasts already draw the sign from the shock, and those paths are unchanged.

What changed

ARCHModel.forecast passes distribution.partial_moment(2, 0) when the volatility model is asymmetric and the method is analytic. GARCH._analytic_forecast and MIDASHyperbolic._analytic_forecast use that weight. The default stays 0.5, so a normal GJR forecast is the same recursion as before.

Verification

Parameters [mu, omega, alpha, gamma, beta, eta, lambda] = [0.05, 0.02, 0.0, 0.20, 0.88, 7.6, -0.16], skew-t, horizon 30:

comparison	max abs difference
analytic vs alpha + k gamma + beta, k = 0.5526	6e-16
analytic vs the old 0.5 recursion	0.15
normal GJR vs the 0.5 recursion	1e-16

Horizon 20, 30,000 simulations: analytic 1.318, simulation 1.317, the old half-moment path 1.11.

pytest arch/tests/univariate/test_variance_forecasting.py -k "midas_analytical or midas_asym or gjr_analytic": 4 passed.

This change was written with AI assistance and checked against the partial-moment recursion and the simulation above.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 99.59%. Comparing base (cc8df46) to head (3e43243).

Additional details and impacted files

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