# 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.