spb/wp46_uqo

1# Model card — `aparch11` (APARCH(1,1), asymmetric power ARCH)23## 1. Identity4- **model_id:** `aparch11` · **family:** power · **tier:** A · **status:** implemented · **impl:** arch 8.0.0 (`APARCH(p=1, o=1, q=1)`, δ estimated)5- **Label:** APARCH(1,1) · **distributions:** normal, t, ged, skewt6- **Seminal citation:** `ding1993long`7- **Other references:** `ling2002necessary` (moment conditions), `he1999properties`, `giot2003value` (skewed-t APARCH for VaR), `brownlees2011practical`, `hansen2005forecast`.89## 2. Definition (arch parameterisation)10$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\qquad \sigma_t^{\delta} = \omega + \alpha\big(|\varepsilon_{t-1}| - \gamma\,\varepsilon_{t-1}\big)^{\delta} + \beta\,\sigma_{t-1}^{\delta} ,$$11`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.1213## 3. Parameter space and admissibility14- Parameters: `mu, omega, alpha[1], gamma[1], beta[1], delta` (+ distribution).15- 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`).16- Admissibility flag: **always `True`** (no model-specific check in `ArchModel.fit`).1718## 4. Nesting and relations19Nests `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".2021## 5. Estimation22arch `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.2324## 6. Forecasting25h = 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).2627## 7. Implementation28- `src/wp46/models/arch_models.py` (`SPECS["aparch11"]`), arch 8.0.0 (`sheppard2025arch`).29- 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).30- 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.3132## 8. Known limitations / failure modes33δ 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.3435## 9. Computational footprint36Most 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).3738## 10. Literature expectations39`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.40