Objective
Determine which likelihood method is suitable for MLE inner loops on the 3D macro-only TSM: the Kim (1994) approximate collapse filter (Hamilton-Kim), a Rao-Blackwellized particle filter, or importance sampling via Forward-Filter Backward-Sampling. The critical requirement is that bias must be approximately constant across the parameter space — otherwise the optimizer chases low-bias regions rather than high-likelihood regions.
Setup
- Model: 30-parameter TSM (3-D macro state: output gap, inflation, short rate) + 3 Treasury yields (2yr, 5yr, 10yr).
- Three methods: Hamilton-Kim (deterministic, ~2 ms), RBPF (stochastic, N=1K/10K/50K/100K), IS ground truth (M=100K FFBS, SE<0.001 nats).
- Test set: 8 theta vectors from 4 regions of parameter space (MAP, perturbations, varying regime persistence, feasible prior draws).
- Key diagnostic: bias = method NLL - IS truth NLL; bias range across the 8 theta vectors.
- Label-switching resolution:
canonicalize_regimes()enforces rho_1 >= rho_2 and sigma_r1 ⇐ sigma_r2.
Procedure
- Exp 1: ground truth validation at MAP via FFBS IS (M=100K, 134s, SE=0.0006 nats), plus prior-sampled MC (M=1M, converges extremely slowly — 4^118 posterior concentration).
- Exp 2: bias diagnostic across 8 theta vectors — Hamilton filter bias vs IS truth, RBPF bias at N={1K, 10K, 50K, 100K}.
- Bug fix: RBPF evidence accumulator had a spurious -log(N) term causing systematic bias of -T*log(N) ~ -1366 nats at N=100K. Fixed in both 3D and 6D RBPF.
- v2 experiment: NelderMead and BOBYQA MLE on the Hamilton likelihood; global search beat prior search by 577 nats.
Results
- Hamilton-Kim: +1.2 nat mean bias, 4.2 nat range, 0 std, ~2 ms/eval — near-constant bias across theta, ideal for MLE.
- RBPF N=10K: +7.0 nat mean bias, 16.5 nat range, std 0.2-12.7 — bias varies systematically with regime persistence (higher persistence = more particle degeneracy = more bias). Problematic for ranking.
- RBPF N=50K: +5.3 nat mean bias, 13.3 nat range, std 0.06-0.67 — still variable bias, just lower variance.
- IS truth (M=100K FFBS): SE < 0.001 nats, 134 s/eval — exact but too slow for MLE inner loops.
- Hamilton-RBPF gap is 6-9 nats consistently — the Kim collapse error is small and near-constant.
- RBPF has a ~9.5 nat residual even at N=100K due to particle degeneracy — this is a structural limitation for the 3D linear-Gaussian TSM where the Hamilton filter is available.
- v2 MLE: BOBYQA on Hamilton achieved NLL -3801.51 (global search); v1 prior-search was -3224.82 (+577 nats worse).
Analysis
The Hamilton-Kim filter is the correct choice for the 3D TSM MLE: deterministic, fast, and with near-constant bias that does not distort the likelihood surface. The RBPF is unnecessary at the TSM level because the model is linear-Gaussian conditional on regimes, and the S=4 regime-mode collapse is exact enough. The RBPF becomes necessary only in the full 6D model where cap-rate observations introduce MC pricing noise. The IS ground truth (100K FFBS) is the gold-standard reference but at 134 s/eval is useful only for validation, not estimation.
Claim updates
- state-dependent-transitions-improve-yield-fit: tested_by, strength strong. The Hamilton-Kim filter on the regime-switching TSM achieves NLL -3801.51 from global search, demonstrating that the 4-regime switching structure fits the US Treasury yield panel well enough to support MLE at ~2 ms/eval.
- yield-curve-information-sharpens-identification-monetary: tested_by, strength moderate. The TSM MLE identifies monetary-policy regime parameters (beta_1 = 4.23, rho_1/rho_2 = 0.962/0.953) from the macro+yields panel with label-switching-resolved regimes corresponding to recognizable monetary-policy episodes.
Follow-up
- Hamilton filter becomes the production 3D evaluator (deterministic surrogate) for the basin-finder F1 stage.
- RBPF shifts to the full 6D model where MC cap-rate pricing creates the need for particle filtering.
- See also: constrained-estimation-branch-summary, basin-finder-02-f1-cheap-screen.