# Model card — msgarch2 (two-regime Markov-switching GARCH, Haas–Mittnik–Paolella)

# 1. Identity

• model_id: msgarch2 · family: regime-switching · tier: A · status: implemented · impl: custom (src/wp46/models/custom.py::MSGARCH2)
• Label: MS-GARCH(2) (Haas-Mittnik-Paolella) · distributions: normal, t (common ν across regimes) · registry note: "regimes ordered by unconditional variance"
• Seminal citation: haas2004new
• Other references: hamilton1989new (filter), gray1996modeling, klaassen2002improving (collapsing MS-GARCH — different likelihoods, not implemented), bauwens2010theory, augustyniak2014maximum (path-dependent MS-GARCH, out of scope), ardia2019markov/ardia2016msgarch (MSGARCH R package implementing the HMP form), ardia2018forecasting, marcucci2005forecasting (benchmarks).

# 2. Definition

Regime $S_t \in {1,2}$, first-order Markov chain with $P(S_t = 1 \mid S_{t-1} = 1) = p_{11}$, $P(S_t = 2 \mid S_{t-1} = 2) = p_{22}$; two parallel GARCH(1,1) variances driven by the common innovation: $$r_t = \mu + \varepsilon_t,\qquad \sigma_{k,t}^2 = \omega_k + \alpha_k,\varepsilon_{t-1}^2 + \beta_k,\sigma_{k,t-1}^2,\ k = 1,2,\qquad \varepsilon_t \mid S_t = k,\ \mathcal F_{t-1} \sim \sigma_{k,t} z_t ,$$ $z_t$ N(0,1) or standardised Student-t with a common ν. hamilton_msgarch: $\sigma_{k,0}^2 = s_0$ (backcast) for both regimes, prior $P(S_0 = 1) = \pi_1 = (1-p_{22})/(2 - p_{11} - p_{22})$ (ergodic distribution), Hamilton filter with log-sum-exp; the log-likelihood is $\sum_t \log\big[\sum_k P(S_t = k\mid\mathcal F_{t-1}),f_k(\varepsilon_t)\big]$. Identification: regime 1 = lower unconditional variance $u_k = \omega_k/(1-\alpha_k-\beta_k)$.

# 3. Parameter space and admissibility

• Parameters: mu, p11, p22, omega1, alpha1, beta1, omega2, alpha2, beta2 (+ nu).
• Reparameterisation: $p_{11}, p_{22} = \sigma(\theta)$; per regime $\omega_k = e^{\theta}$, $\alpha_k + \beta_k = \sigma(\theta) \in (0,1)$, $\alpha_k = (\alpha_k+\beta_k),\sigma(\theta')$ — so each regime is covariance stationary by construction.
• Identification constraint $u_1 \le u_2$ enforced by a soft penalty added to the negative log-likelihood: $50,(\ln u_1 - \ln u_2)$ when $u_1 > u_2$ (the reported loglik = −objective therefore includes the penalty if it is active at the optimum).
• Admissibility flag: $\omega_1, \omega_2 > 0$, $0 < p_{11}, p_{22} < 1$. The HMP stationarity condition for the mixture (spectral radius of the regime-weighted persistence matrix < 1) is not computed; per-regime $\alpha_k + \beta_k < 1$ is sufficient for it.

# 4. Nesting and relations

Collapses to garch11 when the two regimes have identical parameters or when $p_{11} \to 1$ (absorbing). HMP form: regime variances depend only on past $\varepsilon^2$ (no path dependence), unlike Gray/Klaassen (collapsing) or bauwens2010theory (path-dependent) — literature-matrix caveat #6. stgarch (unsupported) is the smooth-transition analogue.

# 5. Estimation

Custom MLE (CustomModel.fit with MSGARCH2._negll): L-BFGS-B on θ (maxiter 2000) from three starts: (p11, p22, ω1, α1, β1, ω2, α2, β2) ∈ {(0.98, 0.97, 0.04v, 0.03, 0.90, 0.30v, 0.10, 0.80), (0.95, 0.95, 0.03v, 0.05, 0.85, 0.30v, 0.15, 0.75), (0.99, 0.99, 0.035v, 0.02, 0.93, 0.15v, 0.08, 0.85)} with $v = \operatorname{var}(r)$; Nelder–Mead polish on failure; μ start = sample mean, ν start 6; backcast $s_0$ arch-style. Point estimates only. filter returns the one-step predictive mixture variance $E[\sigma_t^2 \mid \mathcal F_{t-1}] = \pi_{1,t|t-1}\sigma_{1,t}^2 + (1-\pi_{1,t|t-1})\sigma_{2,t}^2$.

# 6. Forecasting

MSGARCH2.forecast (documented in the class as the HMP recursion): with $\boldsymbol\pi_t = (P(S_t=1\mid\mathcal F_t), P(S_t=2\mid\mathcal F_t))$ from the filter, $\mathbf P$ the transition matrix, and $\sigma_{k,t+1}^2 = \omega_k + \alpha_k\varepsilon_t^2 + \beta_k\sigma_{k,t}^2$, $$\boldsymbol\pi_{t+j|t} = \boldsymbol\pi_t\mathbf P^{j},\qquad \hat\sigma^2_{t+j|t} = \sum_k \pi_{k,t+j|t},\hat\sigma^2_{k,t+j},\qquad \hat\sigma^2_{k,t+j+1} = \omega_k + \alpha_k,\hat\sigma^2_{t+j|t} + \beta_k,\hat\sigma^2_{k,t+j},$$ for $j = 1..22$ (e2f = out[j] feeds both regime recursions). The code treats $E[\varepsilon^2_{t+j}\mid\mathcal F_t]$ as the marginal-probability mixture of the regime variances; whether this coincides with the exact HMP multi-step expression (which conditions the future regime and the regime variances jointly) is not verified here — recorded in _discrepancies.md. Cumulative = sum over j ≤ h. Method: analytic. The forecast object also carries regime_probs ($\boldsymbol\pi_{t+22|t}$) and regime_var.

# 7. Implementation

• src/wp46/models/custom.py::MSGARCH2 (numba hamilton_msgarch).
• Cross-check: none against a library (MSGARCH R package not run; not in results/metrics/rugarch_crosscheck.json).
• Tests: tests/unit/test_custom_models.py::test_msgarch_two_regimes_ordered_and_forecast_positive (converged, $u_1 \le 1.01,u_2$, positive 22-step forecasts), test_no_lookahead_in_forecasts (normal); tests/simulation/test_recovery.py::test_msgarch_recovery_regime_separation (T = 6,000, P = [[0.98, 0.02],[0.03, 0.97]], regimes (0.02, 0.03, 0.90)/(0.3, 0.10, 0.80): recovers $p_{11}, p_{22} > 0.9$ and $u_2/u_1 > 3$).

# 8. Known limitations / failure modes

Multimodal likelihood (label switching handled by the soft penalty, which can bias the reported log-likelihood when active); a regime can degenerate (p_kk → 1 or ω_k → 0) → flat likelihood, slow convergence, boundary optima; common ν only; no stationarity flag for the mixture; the Hamilton filter costs a full pass with two GARCH recursions per likelihood evaluation; multi-step recursion caveat above.

# 9. Computational footprint

Analytic forecasts; 9–10 parameters; Hamilton filter per evaluation (≈ 3× the cost of a GARCH(1,1) evaluation plus more iterations). Not in the pilot; expected to be among the slowest Tier A estimations with figarch11 and aparch11.

# 10. Literature expectations

marcucci2005forecasting: MS-GARCH beats single-regime GARCH at short horizons only on S&P 100; ardia2018forecasting (426 assets): MS-GARCH improves VaR/ES and density forecasts rather than point variance accuracy; haas2004new show the analytic tractability of the parallel-process form. Expectation (H4, H6): gains concentrated at h = 1 in regime transitions and in tail-risk evaluation, with higher failure rates (H5).