Objective
Improve the v1 TSM-only MLE via a global 500K Latin hypercube search in phi-space followed by NelderMead polishing of the top 20 candidates, and characterize the resulting parameter estimates for use as a PMMH prior center.
Setup
- Model: 30-parameter macro-only term structure model (TSM), 4 compound regimes
- Evaluator: Hamilton-Kim filter negative log-likelihood (~2 ms/eval)
- Comparators: v1 MLE at NLL -3730.41 (BlackBoxOptim + L-BFGS)
- Acceptance: likelihood improvement + economically sensible parameters + usable Hessian
Procedure
- 500K Latin hypercube sample in phi-space (4.1 min, 1.35% feasibility)
- NelderMead polish of top 20 candidates (10K iter each, ~6.8 min total)
- Re-polish of best candidate with tighter NelderMead
- Gradient verification (60 evals) + finite-difference Hessian (1950 evals)
- Prior construction and feasibility validation at the MLE center
Results
- Best NLL: -3801.51 (71-nat improvement over v1, likelihood ratio ~10^31)
- Yield RMSE: 2yr 28.6 bp, 5yr 27.4 bp, 10yr 36.3 bp (competitive with literature)
- LHS global search found nothing close to the v1 basin (best LHS: -1682, 2048 nats worse)
- Hessian: 5 negative eigenvalues, condition number 8.4e23, gradient at MLE = Inf (boundary)
- Risk premia (8 params): SE=10 uniformly from eigenvalue clipping — fundamentally unidentified by yield data alone
- TSM prior feasibility: 49.9% (excellent); CRE PIxG feasibility: 0% (no-bubble condition fails)
- Audit: not performed (no separate audit session)
Analysis
The v2 global search succeeded at refining the v1 MLE but the LHS exploration was wasted effort — random sampling cannot find the good region. The sharp three-way split between well-identified (rho, sigma, pi; |t|>5), weakly identified (macro intercepts/slopes; |t|<2), and unidentified (risk premia; fake SEs) parameters is the key structural finding. Risk premia are a flat ridge in yield-data likelihood: v1-to-v2 shifts of 10-100x the parameter value produce only 71 nats. CRE cap rate data is needed to pin them down.
Claim updates
- state-dependent-transitions-improve-yield-fit: MLE achieves 27-36 bp yield RMSE with economically coherent regime story (Taylor principle satisfied in active regime, plausible regime durations), supporting the claim that regime-switching structure improves yield fit; but risk premia unidentified by yields alone.
- forward-solution-unique-fundamental-ree-with-nbc: MLE sits on the feasibility boundary where gradient perturbations hit Inf, consistent with the RE solver’s no-bubble constraint being binding at the optimum.
Follow-up
- Use v2 MLE as TSM prior center for PMMH (49.9% TSM feasibility)
- Risk premia need CRE data to identify — 8 parameters are nuisance for TSM
- Tighten CRE prior to sigma_log=0.3 for PIxG feasibility
- Consider profiling out risk premia for PMMH efficiency