SDID time weights: defaults versus optimized reference

Same samples and default penalty · Under development

Scope

This comparison isolates numerical optimization, holding the default time-weight penalty, intercept, outcomes, donor pools and cohort windows fixed. It does not select a new penalty by cross-validation. The 396 attempted blocks are the same ones used in the prior default analysis: 394 return weights, and two zero-noise blocks remain unavailable. No corpus expansion, unit-weight refitting, new ATT estimates or changes to the original 22-paper collection are included.

The original author implementation is used for defaults and tighter-iteration variants. An independent convex quadratic program solves the same objective. Because the normalized ridge coefficient is only 10^-12, objective accuracy alone can leave substantial uncertainty about the weights. All 11 nonzero-noise Italian opera blocks are therefore refined with an 80-digit active-set solver. The other 383 blocks use the double-precision QP, with separate initialization and objective-gap checks. This is a numerical reference, not a claim that all weights are statistically well determined.

What changes

The median L1 change in the time-weight vector is 2.23^{-4} (on a scale from 0 to 2). 77 of 394 change by more than 0.01 in L1. Support above 10^-6 changes in 41 blocks; weight in the last fifth of observed pre-periods changes by more than one percentage point in 23, and more than five in 6. These are dependent cohort/outcome fits, not 394 independent papers or a representative sample of empirical practice.

The default solver passes the normalized objective-gap check (10^-8) in 134/394 cases; the reference passes in 394/394. The default has 16 time-weight iteration caps. These criteria differ: default stopping tests objective improvement, not a global objective-gap certificate. Tighter author iterations alone do not reliably settle every problem.

method	n	accepted	iteration_caps	median_gap	max_gap
default	394	134	16	1.335e-05	0.00067889
author_tight_sparse	394	152	33	2.000e-08	0.00007585
author_tight_no_sparse	394	114	239	3.920e-06	0.00006752
qp_uniform	394	394	2	0.000e+00	0.00000000
qp_default_start	394	394	0	0.000e+00	0.00000000

Weak regularization and numerical precision

Three opera QPs were sensitive to initialization despite tiny objective gaps. The 80-digit solver resolves them: uniform and default starting weights agree within 10^-45 in L1, with high-precision KKT gaps below 10^-50. Refining all 11 opera blocks also changes two vectors whose double-precision initializers had agreed. Thus initialization agreement alone is not a general accuracy certificate. Outside opera, only double-precision numerical checks are supplied.

Some striking support changes barely affect the donor predictions. For example, S08-P003, cohort column 32, changes from two to 30 periods above 10^-6 while its normalized donor predictions change by about 2.4 × 10^-11 RMS. The extra periods are not evidence of a materially better predictive fit. These comparisons separate the sparsity induced by the default algorithm from the minimizer of its weakly regularized objective.

	fit_id	T0	default_support	reference_support	l1_change	objective_reduction	donor_prediction_rms_change
181	S08-P003–y__d–c032	31	2	30	1.8980253	0e+00	0.00000000
178	S08-P002–y__d–c032	31	4	27	1.7037037	0e+00	0.00000000
182	S08-P003–y__d–c057	56	35	56	1.0072050	2e-08	0.00014561
183	S08-P003–y__d–c059	58	34	58	0.9466630	2e-08	0.00012694
180	S08-P002–y__d–c059	58	55	58	0.6300695	2e-08	0.00014596
179	S08-P002–y__d–c057	56	53	56	0.6250873	3e-08	0.00018179
300	S14-P004–y__d–c032	31	19	19	0.4781106	2e-08	0.00014707
311	S14-P004–y__d–c043	42	12	20	0.4711898	0e+00	0.00005349

Preselected examples

These are the same 13 representative examples chosen before the default fits; no examples were replaced to emphasize large differences. Recent mass means the last ceiling(20% × T0) observed pre-periods. This is not always calendar time. Support counts use weight >10^-6; all zeros and small values remain in the exports.

	paper	T0	default_support	reference_support	l1_change	default_recent_mass	reference_recent_mass
1	Religion and conflict	2	2	2	0.0000	0.6966	0.6966
2	Collectivization	3	2	2	0.0000	0.8768	0.8768
3	Rebel on the Canal	176	154	152	0.3252	0.2494	0.2680
4	ARRA and voting	3	2	2	0.0000	0.8100	0.8100
5	Cap-and-trade	4	3	3	0.0000	0.5634	0.5634
6	Medical implants	5	3	3	0.0001	0.5765	0.5765
7	School construction	2	2	2	0.0000	0.9764	0.9764
9	Curriculum and Ideology	2	2	2	0.0000	0.1907	0.1907
148	Government audits (randomized)	8	2	2	0.0000	0.0000	0.0000
176	Italian opera	56	2	2	0.0000	0.0000	0.0000
206	Birth of a Nation	45	15	15	0.0006	0.3287	0.3287
325	Hospitals / Gift of a Lifetime	10	5	5	0.0001	0.4370	0.4370
387	Credit Markets	20	4	4	0.0000	0.7344	0.7344

Credit Markets and the preselected opera example are essentially unchanged. Rebel on the Canal changes more visibly: support is 154 → 152 of 176 periods, L1 change is about 0.325, and recent mass rises from about 24.9% to 26.8%. This preserves the qualitative contrast between recent concentration, older selected periods and diffuse profiles, while showing that numerical settings can matter greatly for particular outcomes or cohorts.

Inspect and reproduce

• Interactive explorer, all 394 blocks
• Profile PDF
• Paired comparisons
• Optimized reference vectors and diagnostics
• All five base solver variants
• Support sensitivity
• Base numerical analysis / executable source
• High-precision refinement / executable source
• Independent NumPy checks

Run the base Quarto analysis, the precision-refinement Quarto document and this summary in that order. Source input matrices and individual fitted objects are not included in shared downloads. Randomized audits remain separately identified in the original inventory; existing provisional design assessments are unchanged.