# Model card — loghar_rv (log-HAR-RV, direct projections with smearing)

# 1. Identity

• model_id: loghar_rv · family: realized (Tier B) · tier: B · status: implemented · impl: ols (src/wp46/models/baselines.py::HARRV(log=True))
• Label: log-HAR-RV (direct, smearing) · distributions: empirical
• Seminal citation: corsi2009simple (log-HAR variant discussed there)
• Other references: andersen2001distribution/andersen2001distributionstock (log-RV ≈ Gaussian), andersen2003modeling; Duan (1983) smearing estimator: reference to be added (_missing_references.md); clements2021practical, branco2024forecasting.

# 2. Definition

Same regressors as har_rv but built from $\log y_t$ (y floored at 1e-8): $\mathbf x_t = [1, \log y_t, \tfrac15\sum_{i<5}\log y_{t-i}, \tfrac1{22}\sum_{i<22}\log y_{t-i}]$; direct regressions per horizon on the log of the average future variance: $$\log\Big(\frac1h\sum_{k=1}^{h}y_{t+k}\Big) = \mathbf x_t'\boldsymbol\beta^{(h)} + e^{(h)}_{t+h},\qquad \hat f_h = \exp(\mathbf x_t'\hat{\boldsymbol\beta}^{(h)})\cdot \hat S_h,\quad \hat S_h = \frac1{n_h}\sum \exp(\hat e^{(h)})$$ (Duan smearing factor = mean of the exponentiated OLS residuals, one per horizon). No innovation law.

# 3. Parameter space and admissibility

As har_rv (12 OLS coefficients + 3 smearing factors in aux["smear"]); forecasts positive by construction; converged iff ≥ 50 complete rows per horizon.

# 4. Nesting and relations

Log version of har_rv; related to the log-linear Realized GARCH measurement equation and to the Gaussianity of log-RV.

# 5. Estimation

OLS (np.linalg.lstsq) in logs, residual-based smearing factor; no optimizer, no retries, point estimates only.

# 6. Forecasting

As har_rv after back-transformation: $C_h = \max(\hat f_h, 10^{-8}),h$ for h = 1, 5, 22; per-step variances by even allocation of the increments $C_1$, $C_5 - C_1$ (4 days), $C_{22} - C_5$ (17 days), with floors; sums over k ≤ 1, 5, 22 equal the direct forecasts. Method: direct.

# 7. Implementation

• src/wp46/models/baselines.py::HARRV(log=True); make_model("loghar_rv").
• Cross-check: none (deterministic OLS). Tests: none dedicated (the integration test and pilot use har_rv only).

# 8. Known limitations / failure modes

The smearing correction assumes homoskedastic residuals (a single factor per horizon); under heteroskedastic log-residuals the back-transformed forecasts are biased; the log floor 1e-8 affects zero-variance sessions; overlapping h = 22 targets.

# 9. Computational footprint

As har_rv (three least-squares fits per refit, negligible).

# 10. Literature expectations

Log-HAR is often preferred for its approximately Gaussian residuals (andersen2001distribution, corsi2009simple); comparisons in clements2021practical and branco2024forecasting report small differences between level and log HAR under QLIKE once the retransformation bias is corrected. Expectation: close to har_rv, more robust to outliers in levels, slightly worse under MSE at h = 1.