Extensions¶

This page presents additional analyses: IV robustness, Bajari--Ye with tender FE, year-by-year stability, FL definition variants, and network diagnostics.

IV Robustness¶

Balance Tests¶

Pre-determined observables are regressed on the LOO instrument with item, year, and PBU FE. All standardized differences are below 0.03\(\sigma\), consistent with quasi-random FL supply variation.

IV with Item-Group x Year FE¶

Adding item-group \(\times\) year FE absorbs product-market-year shocks and restricts identification to within-cell variation across PBUs. This specification is reported in Appendix B as a consistency check.

Placebo IV¶

Sub-threshold always-losers (firms that never win but participate below 14 tenders) are used as the instrument source. The placebo first stage is weak---an order of magnitude smaller than the main instrument---confirming that the FL screen captures a specific, not generic, bidder behavior.

Bajari--Ye with Tender Fixed Effects¶

Re-estimating the Bajari--Ye first stage with tender \(\times\) item FE absorbs all tender-level shocks (reference price, commodity prices, PBU effects). Only within-tender, cross-firm variation remains. This provides a cleaner test of bid coordination.

Year-by-Year Stability¶

Year-by-year coefficients — **Figure 12.** Year-by-year FL coefficients. The coefficient is stable across years for log prices and log firms.

Stable across time

The FL--price association is not driven by a particular time period. Coefficients are positive and significant in most years, with no clear trend.

FL Definition Variants¶

FL definition robustness — **Figure 13.** FL price coefficient under alternative FL definitions: baseline (win rate = 0), low-win-rate variants, and cross-fitting.

Network Diagnostics¶

Winner HHI Distribution¶

Network HHI — **Figure 14.** Distribution of winner HHI across FL firms. Higher HHI indicates that the FL firm consistently loses to the same winners---a signature of concentrated co-bidding partnerships.

Regime Test: Bid Dispersion¶

The empirical bid spread distribution of FL firms matches the simulated Regime 1 (complementary cover bidding) pattern rather than Regime 2 (coordinated).

Regime boxplot — **Figure 15.** Bid spread comparison across simulated regimes and empirical FL data.