AN-022: Bootstrap CI for the BNE decomposition¶
Intuition (plain-language)
Is exclusion-dominance (above 50%) just a point estimate that could be noise? Cluster-bootstrapping the auctions puts the exclusion share's confidence interval at [64.5%, 86.8%] — the whole interval is above the 50% dominance threshold, so the ordering survives statistical inference, not just the point estimate.
Question¶
The point estimate of the absolute exclusion share is 72.0% in non-pharma and 68.8% in pharma (AN-010). The headline claim of H:exclusion-dominates is that this share exceeds 50% — the dominance threshold below which the lost-discipline and protected-pool offset channels would be roughly equal in absolute magnitude. The point estimate clears this threshold; does the 95% confidence interval clear it too?
Design¶
- Sample: same structural cells as AN-010 — BEC Pregão drop-outs across non-pharma standardized and pharma cells.
- Variation: cluster bootstrap at auction level (B = 500 replicates), respecting within-auction correlation from unobserved heterogeneity (paired with the UH correction in AN-014).
- Specification: per bootstrap replicate, (i) resample auctions with replacement within (period × pharma × type) strata, (ii) refit \(F_c\) in three regimes (losers-only / all-bidders / Turnbull NPMLE — see AN-013), (iii) run BNE MC with 500 auctions per scenario, (iv) compute \(\Delta_{\mathrm{total}}\), share intens., share entry.
- Outcomes: 95% CI on \(\Delta_{\mathrm{total}}\), share intens., share entry, by class × regime.
Results¶
Bootstrap CIs (cluster at auction; B = 500 replicates;
tab_v3_bootstrap_ci.tex):
Non-pharma:
| Regime | Δ total [95% CI] | Share intens. [95% CI] | Share entry [95% CI] |
|---|---|---|---|
| Losers-only | 0.255 [0.197, 0.314] | 74.0 [65.8, 85.8] | 26.0 [14.2, 34.2] |
| All-bidders | 0.234 [0.180, 0.291] | 73.2 [64.5, 86.8] | 26.8 [13.2, 35.5] |
| Turnbull | 0.251 [0.194, 0.312] | 71.5 [64.0, 82.2] | 28.5 [17.8, 36.0] |
Pharma:
| Regime | Δ total [95% CI] | Share intens. [95% CI] | Share entry [95% CI] |
|---|---|---|---|
| Losers-only | 0.360 [0.294, 0.433] | 70.6 [62.5, 85.9] | 29.4 [14.1, 37.5] |
| All-bidders | 0.305 [0.245, 0.370] | 70.0 [61.6, 85.2] | 30.0 [14.8, 38.4] |
| Turnbull | 0.341 [0.272, 0.409] | 69.0 [62.3, 80.1] | 31.0 [19.9, 37.7] |
Bootstrap point vs canonical headline
The per-regime point estimates above are the B=500 cluster-bootstrap
means. They differ by Monte Carlo noise from the canonical all-bidders
headline reported elsewhere on the site (absolute exclusion share
72.0% NP / 68.8% PH, net effect 0.227 NP / 0.309 PH; values.tex).
The canonical points sit comfortably inside each regime's 95% CI; the
bootstrap is used for the interval, not to revise the point estimate.
Interpretation¶
The 95% lower endpoint of the exclusion share exceeds 50% in every cell × regime. The lowest lower-endpoint across all 6 specifications is 61.6% (pharma, all-bidders regime); the highest lower-endpoint is 65.8% (non-pharma, losers-only). Even in the most pessimistic bootstrap percentile across all regime choices, the lost-discipline channel accounts for more than 60% of the absolute decomposition. The exclusion-dominant ordering is robust to sampling variability at the 95% level.
Δ_total CI fully excludes zero in every cell. Across all 6 specifications, the lower endpoint of Δ_total is ≥ 0.180 (non-pharma) or ≥ 0.245 (pharma) — economically large and statistically distinguishable from zero. This is the bootstrap-equivalent of the "set-aside raises prices" claim, evaluated on the structural counterfactual rather than the reduced-form DDR.
Regime sensitivity is within CI overlap. The point estimates across losers-only / all-bidders / Turnbull regimes (74.0 / 73.2 / 71.5 in non-pharma; 70.6 / 70.0 / 69.0 in pharma) all sit inside each other's 95% CIs. The dominance ordering does not change with the winner-censoring treatment.
The lower-endpoint floor is the load-bearing number. A 95% CI that runs [64.5, 86.8] is not a claim that the exclusion share is 65–87%. It is a claim that, given sampling variability and the auction-level cluster correlation, we can reject at the 5% level any null where the share is below 64.5%. The dominance-vs-parity threshold (50%) sits well outside the CI on all 6 specifications.
Confidence: yellow. The bootstrap is the within-project gold standard for inference on the structural quantities. It is yellow rather than green because (i) the bootstrap inherits the maintained IPV-clock restriction from H:ipv-clock-admissible, (ii) the within-auction cluster correlation is bounded by the UH correction rather than estimated unrestrictedly, and (iii) B = 500 replicates is sufficient for percentile CIs but could be larger for tail-quantile precision.
Follow-ups¶
- Bias-corrected accelerated (BCa) intervals: percentile CIs may understate width when the bootstrap distribution is skewed. BCa would give a more conservative interval.
- Subsampling variant: an \(m\)-out-of-\(n\) subsampling bootstrap would be more robust to the joint dependence on UH correction + BNE simulation than the cluster bootstrap.
- Stratification by adherence: the bootstrap currently stratifies by (period × pharma × type). Adding adherence-rate strata would align the bootstrap CI with the realistic fiscal-cost range reported in AN-011.