AN-007: Wild-cluster bootstrap inference on the UTG contrast¶
Economic intuition
The urgent-price gap is estimated across a modest number of purchasing units, and those units differ a lot in size. With few, uneven clusters, ordinary standard errors can overstate precision. The Rademacher wild-cluster bootstrap re-draws the data thousands of times to stress-test the gap. It still rejects a zero administrative-minus-litigated gap, so the contrast is not an artifact of how the standard errors are computed.
Question¶
The administrative-minus-litigated urgent-price contrast is identified from variation across purchasing-buyer-units (PBU). The PBU clusters are few and uneven, the setting in which conventional cluster-robust standard errors are least reliable. Does the gap survive inference that is valid with a small, unbalanced number of clusters?
Design¶
- Sample: urgent winner panel of 61,620 purchase-offer-item observations (BEC group 65, São Paulo pharmaceutical procurement, 2009–2019); both fixed-effects variants are bootstrapped on this base panel.
- Variation: the administrative-minus-litigated urgent-price contrast, sign convention admin-minus-litigated.
- Specification: Rademacher wild-cluster bootstrap, clustering on PBU, because the PBU clusters are few and uneven. Two fixed-effects variants are reported: the preferred item+year+PBU specification and a tighter item-by-year-month specification.
- Comparison: the administrative urgent channel is the closest feasible urgent-procurement comparison, not a random or clean benchmark; it is selected and larger.
Results¶
| Specification | Bootstrap p-value |
|---|---|
| Preferred: item + year + PBU FE | 0.0080 |
| Tighter: item-by-year-month FE | 0.0390 |
Rademacher wild-cluster bootstrap, clustering on PBU. Sign convention: admin-minus-litigated.
Output: v10-causal-mechanism/output/tables/tab_utg_boottest.tex.
Interpretation¶
Confidence: green for the stated inference question. Under the preferred item+year+PBU specification the wild-cluster bootstrap rejects a zero gap at p=0.0080; the tighter item-by-year-month specification still rejects at p=0.0390. The analysis directly addresses the small, uneven PBU-cluster concern and the result survives that stress test. The bootstrap disciplines inference only; it does not address selection into the litigated regime, which is handled separately by the Lee bounds in AN-002.
Follow-ups¶
- Report the bootstrap under alternative weight schemes (e.g., Webb six-point) to confirm the rejection is not specific to the Rademacher weights.
- Cross-reference the selection-bounding evidence on the same contrast in the utg-gap-selection-bounded hypothesis.