Address PR #369 R1 P3: refresh placebo docstring + per-draw τ regression

- _placebo_variance_se docstring step 3 now describes the warm-start
  semantics (was still saying "uniform initialization, fresh start"
  after PR #369 landed the warm-start). Adds Parameters entries for
  init_omega / init_lambda / _placebo_indices.
- test_placebo_se_matches_r now also asserts elementwise match between
  Python's placebo_effects and R_PLACEBO_TAUS from the fixture, so a
  permutation that diverged on a single draw but happened to leave
  sd() unchanged would still trip the regression.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: igerber

diff_diff/synthetic_did.py

@@ -1709,9 +1709,17 @@ def _placebo_variance_se(
 
         1. Randomly sample N₀ control indices (permutation)
         2. Designate last N₁ as pseudo-treated, first (N₀-N₁) as pseudo-controls
-        3. Re-estimate both omega and lambda on the permuted data (from
-           uniform initialization, fresh start), matching R's behavior where
-           ``update.omega=TRUE, update.lambda=TRUE`` are passed via ``opts``
+        3. Re-estimate both omega and lambda on the permuted data with
+           ``update.omega=TRUE, update.lambda=TRUE`` semantics. Per-draw FW
+           is warm-started from the fit-time weights — ω is initialized with
+           ``sum_normalize(init_omega[pseudo_control_idx])`` and λ is
+           initialized with ``init_lambda`` — matching R's
+           ``vcov.R::placebo_se`` ``weights.boot$omega = sum_normalize(
+           weights$omega[ind[1:N0_placebo]])`` warm-start. The global FW
+           optimum is init-independent (strict convexity), but the 100-iter
+           pre-sparsify pass converges to different sparsification patterns
+           under uniform vs warm init on a handful of draws — warm-start
+           closes the resulting sub-percent SE drift against R.
         4. Compute SDID estimate with re-estimated weights
         5. Repeat `replications` times
         6. SE = sqrt((r-1)/r) * sd(estimates)
@@ -1736,6 +1744,21 @@ def _placebo_variance_se(
             Convergence threshold for Frank-Wolfe (for re-estimation).
         replications : int, default=200
             Number of placebo replications.
+        init_omega : np.ndarray, optional
+            Fit-time unit weights used to warm-start per-draw ω FW.
+            Subset to pseudo-controls and renormalized inside the loop;
+            mirrors R's ``weights.boot$omega = sum_normalize(weights$omega[
+            ind[1:N0_placebo]])``. Cold-start (uniform init) when ``None``.
+        init_lambda : np.ndarray, optional
+            Fit-time time weights used to warm-start per-draw λ FW;
+            mirrors R passing ``weights.boot$lambda = weights$lambda``
+            through. Cold-start when ``None``.
+        _placebo_indices : np.ndarray, optional
+            Private R-parity test seam. When provided, each row of shape
+            ``(replications, n_control)`` replaces the per-draw
+            ``rng.permutation(n_control)`` so a Python fit can consume
+            R's exact permutation sequence and produce a bit-identical SE
+            (see ``test_placebo_se_matches_r``).
 
         Returns
         -------

tests/test_methodology_sdid.py

@@ -1608,15 +1608,32 @@ def capture_then_call(*args, **kwargs):
         kwargs = dict(captured["kwargs"])
         kwargs["replications"] = replications
         kwargs["_placebo_indices"] = r_perms
-        se_n, _ = sdid._placebo_variance_se(*captured["args"], **kwargs)
+        se_n, placebo_effects_n = sdid._placebo_variance_se(
+            *captured["args"], **kwargs
+        )
         Y_scale = sdid.results_.zeta_omega / kwargs["zeta_omega"]
         py_se = se_n * Y_scale
+        py_taus = np.asarray(placebo_effects_n) * Y_scale
         # Match R within cross-library FW tolerance (Rust vs R BLAS
         # reductions differ at sub-ULP; 1e-8 absorbs that without
         # masking a real divergence).
         assert abs(py_se - r_se) < 1e-8, (
             f"Python placebo SE {py_se} != R {r_se} (delta {py_se - r_se})"
         )
+        # Per-draw τ regression: equal-SE doesn't imply equal sample, and
+        # the placebo τ vector is user-visible through ``placebo_effects``
+        # and feeds the empirical placebo p-value (synthetic_did.py
+        # around L1164-L1170). Compare elementwise so a permutation that
+        # diverged at a single draw — but happened to leave sd() unchanged
+        # — still trips the regression.
+        r_taus = np.asarray(payload["R_PLACEBO_TAUS"], dtype=float)
+        assert r_taus.shape == py_taus.shape == (replications,), (
+            f"shape mismatch: r {r_taus.shape}, py {py_taus.shape}"
+        )
+        np.testing.assert_allclose(
+            py_taus, r_taus, atol=1e-8, rtol=1e-8,
+            err_msg="Per-draw placebo τ diverges from R despite SE match",
+        )
 
 
 # =============================================================================