# Model card — aparch11 (APARCH(1,1), asymmetric power ARCH)

# 1. Identity

• model_id: aparch11 · family: power · tier: A · status: implemented · impl: arch 8.0.0 (APARCH(p=1, o=1, q=1), δ estimated)
• Label: APARCH(1,1) · distributions: normal, t, ged, skewt
• Seminal citation: ding1993long
• Other references: ling2002necessary (moment conditions), he1999properties, giot2003value (skewed-t APARCH for VaR), brownlees2011practical, hansen2005forecast.

# 2. Definition (arch parameterisation)

$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\qquad \sigma_t^{\delta} = \omega + \alpha\big(|\varepsilon_{t-1}| - \gamma,\varepsilon_{t-1}\big)^{\delta} + \beta,\sigma_{t-1}^{\delta} ,$$ aparch_recursion: pre-sample shock $= \sqrt{\text{backcast}}$ (variance backcast) and $\sigma_0^\delta$ term $= \text{backcast}^{\delta/2}$; $\sigma_t^2 = (\sigma_t^\delta)^{2/\delta}$, clamped to arch's variance bounds. Innovations: normal, t, GED, Hansen skew-t.

# 3. Parameter space and admissibility

• Parameters: mu, omega, alpha[1], gamma[1], beta[1], delta (+ distribution).
• arch bounds: $\omega \in [0, 10v]$ with $v = \max(\overline{|\varepsilon|^{0.5}}, \overline{\varepsilon^2})$, $\alpha \in [0,1]$, $\gamma \in [-0.9997, 0.9997]$, $\beta \in [0,1]$, $\delta \in [0.05, 4]$; linear constraints $\omega, \alpha \ge 0$, $|\gamma| < 0.9997$, $\beta \ge 0$, $0.05 < \delta < 4$, $\alpha + \beta < 1$ (arch's generic persistence constraint, not the Ding–Granger–Engle stationarity condition $\alpha E[(|z| - \gamma z)^\delta] + \beta < 1$, ling2002necessary).
• Admissibility flag: always True (no model-specific check in ArchModel.fit).

# 4. Nesting and relations

Nests garch11 (δ = 2, γ = 0), gjr11 (δ = 2; with $\alpha_{GJR} = \alpha(1-\gamma)^2$, $\gamma_{GJR} = \alpha[(1+\gamma)^2 - (1-\gamma)^2]$), avgarch11 (δ = 1, γ = 0), tarch11 (δ = 1), Higgins–Bera NARCH (γ = 0, free δ). Literature-matrix caveat: "multi-step forecast of h not closed-form when δ ≠ 2".

# 5. Estimation

arch ConstantMean + APARCH(1,1,1) + distribution; SLSQP (maxiter 1000) with the bounds/constraints of §3; starting values from arch's grid α ∈ {0.03, 0.05, 0.08, 0.15} × γ ∈ {−0.5, 0, 0.5} × (α+β) ∈ {0.8, 0.9, 0.95, 0.975, 0.99} × δ ∈ {0.5, 1.2, 1.8}, best Gaussian log-likelihood; variance backcast (0.94ⁱ weights on ε²); ≤ 2 restarts (ArchModel.fit). Point estimates only.

# 6. Forecasting

h = 1 analytic (recursion, then $(\cdot)^{2/\delta}$). h > 1 by simulation (arch raises for analytic APARCH multi-step): 2,000 paths of $\sigma^\delta_{t+k}$ via APARCH._simulate_paths (shock $= z,\sigma$, $(|{\rm shock}| - \gamma,{\rm shock})^\delta$ feeds the next step), $\hat\sigma^2_{t+k|t}$ = path mean of $(\sigma^\delta_{t+k})^{2/\delta}$. Normal: seeded (46 + origin); t/GED/skew-t: arch's unseeded fitted-distribution simulator (_discrepancies.md). Cumulative = sum over k ≤ h. Method: simulation (h > 1).

# 7. Implementation

• src/wp46/models/arch_models.py (SPECS["aparch11"]), arch 8.0.0 (sheppard2025arch).
• Cross-check (results/metrics/rugarch_crosscheck.json, rugarch apARCH, t): −3372.7593 (ours) vs −3372.8489 (rugarch), difference +0.0896 (ours higher; δ = 1.81, γ = 0.232, ν = 7.02 on GJR-t data).
• Tests: tests/replication/test_rugarch.py (fit recorded in the JSON); pilot aparch11__t: 207/207 refits ok, 1,032/1,032 forecasts ok on CL.

# 8. Known limitations / failure modes

δ and γ are jointly weakly identified (flat likelihood ridges; δ can run to the bounds 0.05/4 or the persistence constraint binds); γ bound ±0.9997; simulation forecasts with Monte-Carlo error, unseeded for non-normal laws; no admissibility flag beyond optimizer constraints; stationarity under the estimated (γ, δ) is not verified.

# 9. Computational footprint

Most expensive arch-backed estimation (6–8 parameters, 180 grid starting values evaluated before SLSQP): pilot median fit time per refit (t) 0.0321 s (CL) / 0.0406 s (EURUSD), max 0.07 s; forecasts by simulation (22 × 2,000 draws per origin).

# 10. Literature expectations

ding1993long found δ ≈ 1.4 on S&P 500 with strong asymmetry; hansen2005forecast report APARCH among the best on IBM; giot2003value favour skewed-t APARCH for VaR; brownlees2011practical include APARCH among the top asymmetric models. Expectation: competitive with GJR/EGARCH at h = 1 on equities (H1) at the highest estimation and forecasting cost of Tier A (H5), with the largest failure/boundary rate.