# Model card — avgarch11 (AVGARCH(1,1), Taylor–Schwert absolute-value GARCH)

# 1. Identity

• model_id: avgarch11 · family: power · tier: A · status: implemented · impl: arch 8.0.0 (GARCH(p=1, o=0, q=1, power=1.0), arch name "AVGARCH")
• Label: AVGARCH(1,1) (Taylor-Schwert) · distributions: normal, t, ged, skewt
• Seminal citation: taylor2008modelling (2nd ed. of Taylor 1986; the 1986 first edition has no verified record — see docs/literature_matrix.md, Unverified) and schwert1989why.
• Other references: ding1993long (APARCH nests it, δ = 1, γ = 0), zakoian1994threshold.

# 2. Definition (arch, power = 1)

$$r_t = \mu + \varepsilon_t,\ \varepsilon_t = \sigma_t z_t,\qquad \sigma_t = \omega + \alpha,|\varepsilon_{t-1}| + \beta,\sigma_{t-1} ,$$ pre-sample $|\varepsilon|$ and $\sigma$ replaced by the power-1 backcast ($\sum_{i<75} w_i|\varepsilon_i|$, $w_i \propto 0.94^i$). Innovations: normal, t, GED, Hansen skew-t.

# 3. Parameter space and admissibility

• Parameters: mu, omega, alpha[1], beta[1] (+ distribution).
• arch bounds/constraints: $\omega \in [10^{-8}v_1, 10v_1]$ ($v_1 = \overline{|\varepsilon|}$), $\alpha, \beta \in [0,1]$, $\omega > 0$, $\alpha, \beta \ge 0$, $\alpha + \beta < 1$ (arch's generic constraint; the stationarity of $\sigma_t$ requires $\alpha E|z| + \beta < 1$, which is implied).
• Admissibility flag: always True (no check in ArchModel.fit for avgarch11).

# 4. Nesting and relations

aparch11 with δ = 1, γ = 0; tarch11 with γ = 0. Not nested in garch11 (power 1 vs 2). Symmetric; the only symmetric power-1 model of the registry.

# 5. Estimation

arch SLSQP (maxiter 1000), grid starting values, power-1 backcast, ≤ 2 restarts (ArchModel.fit). Point estimates only.

# 6. Forecasting

h = 1 analytic ($\hat\sigma_{t+1|t}^2 = (\omega + \alpha|\varepsilon_t| + \beta\sigma_t)^2$); h > 1 by simulation (2,000 paths of $\sigma_{t+k}$ via GARCH._simulate_paths, power 1; variance forecast = path mean of $\sigma_{t+k}^2$; note $E[\sigma_{t+k}^2] \ne (E\sigma_{t+k})^2$, which is why no analytic rule is used). Normal: seeded (46 + origin); t/GED/skew-t: arch's unseeded distribution simulator (_discrepancies.md). Cumulative = sum over k ≤ h. Method: simulation for h > 1.

# 7. Implementation

• src/wp46/models/arch_models.py (SPECS["avgarch11"]), arch 8.0.0.
• Cross-check: none (not in the rugarch fixture script; rugarch fGARCH(AVGARCH) was not run).
• Tests: none dedicated.

# 8. Known limitations / failure modes

No admissibility flag beyond optimizer constraints; simulation forecasts (Monte-Carlo error, unseeded for non-normal laws); boundary optima as for GARCH(1,1).

# 9. Computational footprint

Estimation like garch11; simulation forecasts (22 × 2,000 draws per origin). No pilot timing.

# 10. Literature expectations

taylor2008modelling/schwert1989why introduced absolute-value dynamics as more robust to outliers; ding1993long found δ ≈ 1.4 for S&P 500 (between AVGARCH and GARCH); hansen2005forecast include TS-GARCH variants without finding them superior to GARCH(1,1). Expectation: similar to GARCH(1,1), possibly more robust in stressed regimes (H6), at a higher forecasting cost (H5).