# Model card — `ewma094` (EWMA / RiskMetrics 1994, λ = 0.94)

## 1. Identity
- **model_id:** `ewma094` · **family:** baseline · **tier:** A · **status:** implemented · **impl:** arch 8.0.0 (`EWMAVariance(lam=0.94)`, λ fixed)
- **Label:** EWMA (RiskMetrics 1994, lambda = 0.94) · **distributions:** normal
- **Seminal citation:** `jpmorgan1996riskmetrics` (RiskMetrics Technical Document, 4th ed.)
- **Other references:** `engle1986modelling` (IGARCH, of which EWMA is the zero-intercept case; registry `nested_in: igarch11`); `zumbach2009riskmetrics` (RM2006 successor).

## 2. Definition
$$r_t = \mu + \varepsilon_t,\qquad \sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1-\lambda)\,\varepsilon_{t-1}^2,\qquad \lambda = 0.94 .$$
arch's `ewma_recursion` starts from $\sigma_0^2$ = backcast. Innovation law: N(0,1) (only μ is estimated; the Gaussian likelihood is used to estimate μ).

## 3. Parameter space and admissibility
- Estimated: `mu` only. λ is fixed at 0.94 (`parameter_names()` is empty for the volatility process).
- $0 < \lambda < 1$ enforced at construction. No stationarity: the process is integrated ($\lambda + (1-\lambda) = 1$) and has no unconditional variance. Admissibility flag always `True` (no check in `ArchModel.fit`).

## 4. Nesting and relations
EWMA = IGARCH(1,1) with $\omega = 0$, $\alpha = 1-\lambda = 0.06$, $\beta = \lambda$ (`igarch11` card); = GARCH(1,1) on the unit-persistence boundary with zero intercept. `rm2006` is a weighted mixture of 14 EWMAs with different decays.

## 5. Estimation
arch `ConstantMean` + `EWMAVariance(0.94)` + `Normal`, `fit(disp="off", options={"maxiter": 1000})`: SLSQP over the single parameter μ (Gaussian log-likelihood with the EWMA variance path); starting value = sample mean; backcast = exponentially weighted average of the first min(75, T) squared residuals (weights ∝ 0.94ⁱ). Restarts (≤ 2) as for all arch-backed models (`src/wp46/models/arch_models.py::ArchModel.fit`). Only point estimates are used in the rolling benchmark; filtering with `am.fix(params)`.

## 6. Forecasting
arch `_analytic_forecast` for EWMA: $\hat\sigma^2_{t+1|t} = \lambda\sigma_t^2 + (1-\lambda)\varepsilon_t^2$ and $\hat\sigma^2_{t+k|t} = \hat\sigma^2_{t+1|t}$ for $k = 2..22$ (flat; a martingale variance). Cumulative $h$-day variance $= h\,\hat\sigma^2_{t+1|t}$. Method: `analytic`.

## 7. Implementation
- `src/wp46/models/arch_models.py` (`SPECS["ewma094"]`, `fixed=True`), arch 8.0.0 (`sheppard2025arch`).
- Cross-check: none in `results/metrics/rugarch_crosscheck.json` (parameter-free recursion; rugarch has no fixed-λ EWMA entry in the fixture script).
- Tests: no dedicated test; not in the pilot.

## 8. Known limitations / failure modes
Fixed decay calibrated on 1990s daily data; integrated dynamics give a flat term structure (no mean reversion), which the literature identifies as its main weakness at h = 5, 22. Cannot fail to converge in the variance recursion (only μ is optimised).

## 9. Computational footprint
One-parameter SLSQP, analytic forecasts: among the cheapest estimated models. No pilot timing.

## 10. Literature expectations
`hansen2005forecast` include RiskMetrics-type EWMA among the dominated benchmarks on the DM/$ and IBM series; `jpmorgan1996riskmetrics` is a risk-management standard rather than a forecasting contender. The literature matrix (§a.1) lists it as an "industry baseline; flat term structure" — expectation: close to GARCH(1,1) at h = 1 in calm periods, inferior at h = 22 and in regimes where mean reversion matters.