# Model card — egarch11 (EGARCH(1,1))

# 1. Identity

• model_id: egarch11 · family: asymmetric · tier: A · status: implemented · impl: arch 8.0.0 (EGARCH(p=1, o=1, q=1))
• Label: EGARCH(1,1) · distributions: normal, t, ged, skewt
• Seminal citation: nelson1991conditional
• Other references: straumann2006quasi (EGARCH QMLE theory), harvey2013dynamic/harvey2014egarch (Beta-t-EGARCH, score-driven relative), hansen2005forecast, brownlees2011practical.

# 2. Definition (arch parameterisation)

$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\ e_t = \varepsilon_t/\sigma_t,\qquad \ln\sigma_t^2 = \omega + \alpha\big(|e_{t-1}| - \sqrt{2/\pi}\big) + \gamma,e_{t-1} + \beta,\ln\sigma_{t-1}^2 .$$ Here arch's alpha[1] multiplies the magnitude term and gamma[1] the sign term (Nelson writes $g(z) = \theta z + \gamma(|z| - E|z|)$; rugarch writes α on the sign and γ on the magnitude — the cross-check confirms the swap: our α = 0.1565 ↔ rugarch γ₁ = 0.1565, our γ = −0.0444 ↔ rugarch α₁ = −0.0444). The centring constant is $\sqrt{2/\pi} = E|z|$ for the normal only; for t/GED/skew-t $E|z| \ne \sqrt{2/\pi}$ and the difference $\alpha(E|z| - \sqrt{2/\pi})$ is absorbed by ω (observationally equivalent, but ω is not Nelson's ω). Pre-sample: $\ln\sigma_0^2 = \omega + \beta\cdot\ln(\text{backcast})$ (the α, γ terms are omitted at t = 0). $\ln\sigma_t^2$ is capped at LNSIGMA_MAX and $\sigma_t^2$ 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] (+ distribution).
• arch bounds: $\omega \in [\ln v - \ln 10^4, \ln v + \ln 10^4]$ ($v = \overline{\varepsilon^2}$), $\alpha, \gamma \in (-\infty, \infty)$, $\beta \in [0, 1]$; constraint $\beta < 1$. arch therefore excludes $\beta < 0$ (Nelson's condition is $|\beta| < 1$).
• Admissibility flag (ArchModel.fit): $|\beta| < 1$ (log-variance stationarity). No positivity constraints are needed (log specification). Note the literature matrix caveat: stationarity is $|\beta| < 1$, not $\alpha + \beta < 1$.

# 4. Nesting and relations

Does not nest garch11 (log vs level). γ = 0 gives a symmetric log-GARCH; the Beta-t-EGARCH of Harvey is the score-driven counterpart (not in the registry; gas_t11 is the variance-level GAS). Exponential form of the asymmetry shared with realegarch11 (unsupported).

# 5. Estimation

arch ConstantMean + EGARCH(1,1,1) + distribution; SLSQP (maxiter 1000) with the bounds of §3; starting values from arch's grid (α ∈ {0.01, 0.05, 0.1, 0.2}, γ ∈ {−0.1, 0, 0.1}, β ∈ {0.5, 0.7, 0.9, 0.98}, ω = (1−β)·ln v̄; best Gaussian log-likelihood); backcast = log of the exponentially weighted mean of the first min(75, T) squared residuals; ≤ 2 restarts (ArchModel.fit). Point estimates only; daily filtering with am.fix(params).

# 6. Forecasting

h = 1: analytic, $\hat\sigma^2_{t+1|t} = \exp(\omega + \alpha(|e_t| - \sqrt{2/\pi}) + \gamma e_t + \beta\ln\sigma_t^2)$. h > 1: simulation (SPECS["egarch11"]["analytic"] = False; arch has no analytic EGARCH multi-step): 2,000 paths, $\ln\sigma^2_{t+k} = \omega + \alpha(|z_{t+k-1}| - \sqrt{2/\pi}) + \gamma z_{t+k-1} + \beta\ln\sigma^2_{t+k-1}$ with simulated standardised shocks, $\hat\sigma^2_{t+k|t}$ = path mean of $\exp(\ln\sigma^2_{t+k})$. Normal: seeded default_rng(46 + origin).standard_normal; t/GED/skew-t: arch's fitted-distribution simulator, unseeded (_discrepancies.md). Cumulative = sum over k ≤ h. Method: simulation (h > 1).

# 7. Implementation

• src/wp46/models/arch_models.py (SPECS["egarch11"]), arch 8.0.0 (sheppard2025arch).
• Cross-check (results/metrics/rugarch_crosscheck.json, rugarch eGARCH, t): −3376.6704 (ours) vs −3376.6833 (rugarch), difference +0.0129; coefficients agree after the α/γ label swap (β 0.96868 vs 0.96871, ν 6.98 vs 6.99); ω differs (−0.0110 vs −0.0171) because rugarch uses the distribution-specific $E|z|$ in the centring (observational equivalence through ω).
• Tests: tests/replication/test_rugarch.py (fit recorded; no assertion on EGARCH beyond the JSON); pilot egarch11__t 1,032/1,032 forecasts ok on CL, 207/207 refits ok.

# 8. Known limitations / failure modes

β frequently close to 1 (log-variance near unit root) → explosive multi-step simulations are bounded only by LNSIGMA_MAX/variance bounds; multi-step forecasts carry Monte-Carlo error and, for non-normal laws, are unseeded; $E[\sigma^2_{t+k}]$ is a mean of log-normal-like paths and is sensitive to tail draws (large variance of the path mean at h = 22); the centring constant mismatch for non-normal laws is harmless for the fit but makes omega non-comparable across distributions.

# 9. Computational footprint

Estimation cheap (pilot median fit time per refit, t: 0.0088 s CL / 0.0095 s EURUSD); forecasting by simulation (22 × 2,000 draws per origin) is the dominant cost.

# 10. Literature expectations

hansen2005forecast: EGARCH-type leverage models among those that beat GARCH(1,1) on IBM; brownlees2011practical rank EGARCH/asymmetric models best for US equities; nelson1991conditional originally with GED innovations. Expectation: strong at h = 1 on equities (H1), weaker at h = 22 where the simulation mean of exponentials inflates forecasts; no gain expected on FX (hansen2005forecast).