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AN-032: Matched-heterogeneity audit — does the quadrant profile survive PS matching?

Intuition (plain-language)

An honest negative. The eye-catching cell-level heterogeneity in the cobidder profile (e.g. Low-HHI × Low-pairs +0.09) mostly evaporates once firms are matched on participation volume (→ +0.07, not significant). Translation: much of what looked like a structured "cartel geography" was really just volume showing up in different cells. What does survive matching is the bid-level conduct (AN-031), so the paper keeps the conduct claim and drops the cell-geography claim.

Question

Does the quadrant-level heterogeneity of the cobidder profile (across HHI × pairs cells) survive propensity-score matching, or is it a volume-confound artifact? The unified-mechanism profile (AN-024) shows cell-level coefficients; this audit asks whether those cells maintain their signal once volume is matched away.

Design

  • Sample: always-loser firms partitioned by:
  • HHI (high / low): product-portfolio concentration.
  • Pairs (high / low): cobidder-pair density.
  • Yielding 4 quadrants.
  • Match: propensity-score matching within each quadrant on participation history (primarily tenders_count).
  • Outcome: cobidder rate proxy at the quadrant level (or related treatment effect; details in scripts/32_matched_heterogeneity.R).
  • Statistics: coefficient, SE, p-value, N, n_FL for both unmatched ATE and PS-matched ATT.

Results

Quadrant Sample Coef SE p N n_FL
Low HHI × Low pairs unmatched +0.090 0.046 0.048 5,294 1,172
Low HHI × Low pairs matched +0.066 0.054 0.227 1,203 1,127
Low HHI × High pairs unmatched −0.113 0.084 0.183 899 448
Low HHI × High pairs matched −0.073 0.111 0.511 361 439
High HHI × Low pairs unmatched +0.266 0.119 0.026 2,147 282
High HHI × Low pairs matched +0.197 0.115 0.091 292 267
High HHI × High pairs unmatched +0.503 0.465 0.290 71 69

Source: output/matched_heterogeneity/matched_het_results.csv.

Matched-heterogeneity audit: unmatched vs matched quadrant coefficients

Figure: quadrant coefficients (HHI × pairs) under unmatched ATE (blue) vs PS-matched ATT (orange). Low-HHI × Low-pairs cell drops from +0.090 (p=0.048) to +0.066 (p=0.227). High-HHI × Low-pairs drops from +0.266 (p=0.026) to +0.197 (p=0.091). Quadrant heterogeneity largely does not survive matching — the honest negative finding for the causal-mechanistic reading of H5.

Attenuation pattern

Quadrant Unmatched Matched Attenuation Loses significance?
Low HHI × Low pairs (largest cell) +0.090** +0.066 27% drop Yes (0.048 → 0.227)
Low HHI × High pairs −0.113 −0.073 35% drop No (was already n.s.)
High HHI × Low pairs +0.266** +0.197 26% drop Marginal (0.026 → 0.091)
High HHI × High pairs +0.503 (small N) — (was already n.s.)

The two cells that reached unmatched significance (Low-Low and High-Low) both lose or nearly lose significance under matching.

Interpretation

The matched-heterogeneity audit is the honest negative finding for H5. Three readings:

  1. Quadrant heterogeneity mostly does not survive matching. The significant cells in the unmatched ATE specification (Low-HHI × Low-pairs and High-HHI × Low-pairs) attenuate by 26–27% and either lose (Low-Low: p 0.048 → 0.227) or barely retain (High-Low: p 0.026 → 0.091) statistical significance under PS matching. The mass cell (Low-Low, N = 5,294 unmatched) loses its significance entirely.

  2. The within-FL profile is largely a volume effect. Cobidders ARE distinct from non-cobidder FLs descriptively (AN-008, AN-028), but the distinctness is concentrated in dimensions correlated with tenders_count. Matching on volume removes the bulk of the within-FL coefficient. This parallels the AN-021 first-time-FL channel demoting from +0.10 unconditional to +0.06 under PS matching.

  3. What survives matching is the bid-level conduct (AN-031). The median gap-to-winner d = −0.28 is at the bid level, not the firm-volume level; it is harder to attenuate by matching on firm-level covariates. This is the remaining behavioral signature when volume is controlled.

For H:cobidder-profile-distinct, the implications are:

  • Descriptive distinctness: robust (AN-008, AN-009, AN-028, AN-031).
  • Causal/mechanistic distinctness: largely a volume effect at the quadrant level; the bid-level conduct remains as the smallest surviving channel.
  • The reading is Mixed → Partial (strongly supported) rather than Confirmed: descriptive evidence is strong, but the matched-attenuation is the honest boundary.

The page documents the failure for transparency, in keeping with mr-frequent's "report the negative" rule.

Follow-ups

  • Re-estimate the bid-level patterns (AN-031) under the same matched specifications — does the bid-conduct distinctness survive matching where the quadrant heterogeneity does not?
  • Decomposition: which volume-correlated variables drive the attenuation? (tenders_count alone? buyer breadth?)
  • Sensitivity to match quality (caliper, common support).
  • Add macros \valMatchedHetLLOne (low-low matched) through \valMatchedHetHHOne to the scripts/99_make_paper_values.R pipeline.