# Model card — garch21 (GARCH with two ARCH lags: arch GARCH(p=2, o=0, q=1))

# 1. Identity

• model_id: garch21 · family: classical · tier: A · status: implemented · impl: arch 8.0.0 (GARCH(p=2, o=0, q=1))
• Label: GARCH(2,1) · distributions: normal, t, ged, skewt
• Seminal citation: bollerslev1986generalized
• Other references: nelson1992inequality (positivity for higher orders), bougerol1992stationarity (strict stationarity of GARCH(p,q)), francq2004maximum (QMLE for general orders), bollerslev2010glossary.
• Naming caveat: in arch, p is the number of lagged squared residuals and q the number of lagged variances, so this is $\sigma_t^2 = \omega + \alpha_1\varepsilon_{t-1}^2 + \alpha_2\varepsilon_{t-2}^2 + \beta_1\sigma_{t-1}^2$. In Bollerslev's (1986) convention (p = GARCH lags, q = ARCH lags) the same model is written GARCH(1,2). The registry label follows arch's convention (recorded in _discrepancies.md).

# 2. Definition

$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\qquad \sigma_t^2 = \omega + \alpha_1\varepsilon_{t-1}^2 + \alpha_2\varepsilon_{t-2}^2 + \beta_1\sigma_{t-1}^2 ,$$ pre-sample $\varepsilon^2$ and $\sigma^2$ replaced by the backcast. Innovations: normal, t(ν), GED(ν), Hansen skew-t(η, λ).

# 3. Parameter space and admissibility

• Parameters: mu, omega, alpha[1], alpha[2], beta[1] (+ distribution).
• arch bounds/constraints: $\omega \in [10^{-8}v, 10v]$, $\alpha_i, \beta_1 \in [0,1]$, $\omega > 0$, $\alpha_i \ge 0$, $\beta_1 \ge 0$, $\alpha_1 + \alpha_2 + \beta_1 < 1$. The non-negativity constraints are sufficient but not necessary for positivity when p > 1 (nelson1992inequality): arch excludes admissible negative-$\alpha_2$ solutions.
• Admissibility flag: $\alpha_1 + \alpha_2 + \beta_1 < 1$.

# 4. Nesting and relations

Nests garch11 ($\alpha_2 = 0$) and ARCH(2). Pre-registered order variant (protocol §4: "(1,1) everywhere, plus GARCH(2,1)/(1,2) and ARCH(5)"). Not nested in garch12.

# 5. Estimation

As garch11 (arch SLSQP, grid starting values with $\alpha$ split equally across the two ARCH lags, backcast initialisation, ≤ 2 restarts; src/wp46/models/arch_models.py::ArchModel.fit). Only point estimates are used in the rolling benchmark.

# 6. Forecasting

arch GARCH._analytic_forecast: $\hat\sigma^2_{t+k|t} = \omega + \alpha_1\widetilde{\varepsilon^2}{t+k-1} + \alpha_2\widetilde{\varepsilon^2}{t+k-2} + \beta_1\tilde\sigma^2_{t+k-1}$ with observed $\varepsilon^2_s$, $\sigma^2_s$ for $s \le t$ and their forecasts for $s > t$ ($E[z^2] = 1$). Cumulative = sum over k ≤ h. Method: analytic.

# 7. Implementation

• src/wp46/models/arch_models.py (SPECS["garch21"]), arch 8.0.0.
• Cross-check: not in the rugarch fixture (only the (1,1) orders are cross-checked); the recursion is the same garch_recursion validated for (1,1) in tests/unit/test_custom_models.py.
• Tests: none dedicated.

# 8. Known limitations / failure modes

Weak identification of $\alpha_2$ (often at the zero bound → SLSQP boundary convergence); near-integrated persistence as for GARCH(1,1); extra parameter rarely improves out-of-sample (expectation §10).

# 9. Computational footprint

Analytic; one more parameter than garch11. No pilot timing.

# 10. Literature expectations

hansen2005forecast found higher-order GARCH specifications no better than (1,1) out of sample; bollerslev1986generalized and the surveys (bollerslev1992arch) note that (1,1) is adequate for most daily series. Expectation: no gain over garch11, slightly more failures.