# Model card — nagarch11 (NAGARCH(1,1), Engle–Ng nonlinear asymmetric GARCH)

# 1. Identity

• model_id: nagarch11 · family: asymmetric · tier: A · status: implemented · impl: custom (src/wp46/models/custom.py::NAGARCH11)
• Label: NAGARCH(1,1) (Engle-Ng) · distributions: normal, t, ged, skewt
• Seminal citation: engle1993measuring
• Other references: higgins1992class (NARCH — a different "NGARCH", literature-matrix caveat #3), glosten1993relation, sentana1995quadratic (QGARCH, the other shifted/linear-asymmetry form), brownlees2011practical (NGARCH in their comparison).

# 2. Definition

$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\qquad \sigma_t^2 = \omega + \alpha\big(\varepsilon_{t-1} - \theta,\sigma_{t-1}\big)^2 + \beta,\sigma_{t-1}^2 ,$$ rec_nagarch with $\sigma_0^2 = \omega + [\alpha(1+\theta^2) + \beta],s_0$ ($s_0$ = backcast). Equivalent to rugarch's fGARCH/NAGARCH $\sigma_t^2 = \omega + \alpha\sigma_{t-1}^2(z_{t-1} - \eta)^2 + \beta\sigma_{t-1}^2$ with $\eta = \theta$. Innovations: normal, t, GED, Hansen skew-t.

# 3. Parameter space and admissibility

• Parameters: mu, omega, alpha, theta, beta (+ distribution).
• Reparameterisation: $\omega = e^{\theta_0}$; persistence $s = \alpha(1+\theta^2) + \beta = \sigma(\theta_1) \in (0,1)$; share $w = \alpha(1+\theta^2)/s = \sigma(\theta_2)$; $\theta$ unconstrained. Hence $\alpha = s w/(1+\theta^2)$, $\beta = s(1-w)$.
• Admissibility flag: $\omega > 0$, $\alpha, \beta \ge 0$, $\alpha(1+\theta^2) + \beta < 1$ (covariance stationarity under $E z = 0$, $E z^2 = 1$ — holds for all four laws, including Hansen's skew-t which is standardised to zero mean and unit variance).

# 4. Nesting and relations

Nests garch11 (θ = 0). News-impact curve = GARCH parabola shifted to the right by $\theta\sigma_{t-1}$ (asymmetry via a shift, not a kink as in gjr11); QGARCH is the linear-term version ($\gamma\varepsilon_{t-1}$ instead of the shift, same quadratic form with $\gamma = -2\alpha\theta\sigma_{t-1}$ varying over time).

# 5. Estimation

Custom MLE (CustomModel.fit): L-BFGS-B on θ (maxiter 2000, ftol 1e-10, gtol 1e-6) from three starts (ω, α, θ, β) ∈ {(0.05v, 0.05, 0.5, 0.88), (0.02v, 0.08, 1.0, 0.85), (0.1v, 0.1, 0.0, 0.8)}, Nelder–Mead polish on failure; μ start = sample mean, ν/η/λ starts 6 / 1.5 / (6, −0.05); backcast $s_0$ = arch-style exponentially weighted mean of the first 75 squared demeaned returns. Point estimates only.

# 6. Forecasting

NAGARCH11.step_forecast: $\hat\sigma^2_{t+1|t} = \omega + \alpha(\varepsilon_t - \theta\sigma_t)^2 + \beta\sigma_t^2$; for $k \ge 2$, $$\hat\sigma^2_{t+k|t} = \omega + \big[\alpha(1+\theta^2) + \beta\big],\hat\sigma^2_{t+k-1|t},$$ exact because $E[(\varepsilon - \theta\sigma)^2 \mid \sigma^2] = \sigma^2(1 + \theta^2)$ when $E z = 0$, $E z^2 = 1$. Cumulative = sum over k ≤ h. Method: analytic.

# 7. Implementation

• src/wp46/models/custom.py::NAGARCH11 (numba rec_nagarch).
• Cross-check (results/metrics/rugarch_crosscheck.json, rugarch fGARCH submodel NAGARCH, t): −3373.3123 (ours) vs −3373.3505 (rugarch), difference +0.0382.
• Tests: tests/replication/test_rugarch.py (assert ours ≥ rugarch − 1); tests/unit/test_custom_models.py::test_nesting_and_admissibility[NAGARCH11] (log-likelihood ≥ GARCH(1,1) − 0.5, admissible, positive 22-step forecasts), test_no_lookahead_in_forecasts; tests/simulation/test_recovery.py::test_nagarch_recovery (T = 5,000, normal, θ = 0.8 recovered within 0.3, β within 0.05); pilot nagarch11__t: 207/207 refits ok on CL.

# 8. Known limitations / failure modes

θ is weakly identified when α is small (the shift multiplies α); large |θ| with tiny α yields flat likelihoods and slow L-BFGS-B progress (pilot max fit time 0.17 s on CL vs median 0.009 s). Near-integrated persistence as in GARCH. The sigmoid reparameterisation cannot return exactly $s = 1$ (inadmissible optima can only arise from numerical failure).

# 9. Computational footprint

Analytic; 5–7 parameters. Pilot median fit time per refit (t): 0.0089 s (CL), 0.0218 s (EURUSD).

# 10. Literature expectations

engle1993measuring show the NAGARCH news-impact curve fits Japanese equity data better than GARCH; brownlees2011practical include NGARCH among the best performers on US equities; hansen2005forecast find leverage specifications superior on IBM only. Expectation: comparable to GJR/EGARCH at h = 1 on equities (H1); exact analytic multi-step forecasts are an advantage over the simulation-based asymmetric models at h = 22 (H5).