Robustness¶

We stress-test the screen from every angle---definition, data, assumptions, and alternative interpretations.

1. FL Definition Robustness¶

IQR Threshold Sensitivity¶

The coefficient decreases monotonically with stricter thresholds, most likely reflecting classification noise as the treated group shrinks. A two-dimensional sensitivity analysis varying both the IQR multiplier and win-rate cutoff produces significant coefficients across all 36 cells.

Multiplier	FL Firms	Coefficient	SE
1.0x	3,442	0.079	(0.023)
1.5x (baseline)	2,735	0.064	(0.020)
2.0x	2,093	0.060	(0.022)
3.0x	1,456	0.050	(0.024)

Threshold stability — **Figure 8.** FL price coefficient across IQR multiplier thresholds. Error bars show 95% confidence intervals.

Cross-Fitting¶

FL firms defined using odd years, regressions estimated on even years (and vice versa). The cross-fold average (0.036) is attenuated relative to full-sample OLS (0.064) but remains positive. A decomposition exercise yields 0.043 (SE = 0.019), indicating roughly two-thirds of the attenuation reflects classification noise from the smaller subsample, not mechanical bias.

Continuous Treatment¶

A continuous measure---log of the most active always-loser's participation count---yields 0.022 (SE = 0.005, \(p < 0.01\)) per log-point.

2. Sample and Specification Robustness¶

Check	Coefficient	\(N\)	Note
OLS (item + year + PBU FE)	0.064	1,654,401	Baseline
CEM matching	0.077	969,751	Coarsened exact matching
IPW matching	0.055	830,194	Inverse probability weighting
Item x year FE	0.074	1,654,401	Tighter controls
Two-way clustering (item + PBU)	0.064 (SE = 0.024)	1,654,401	Significant under all clustering
Horse-race (FL + Imhof CV)	0.084	1,654,401	Suppression effect
Decomposition (odd-yr FL)	0.043	1,654,401	Intermediate

3. Sensitivity to Unobservables (Cinelli--Hazlett)¶

**Figure 9.** Sensitivity contour plot (Cinelli & Hazlett, 2020). The robustness value RV = 17.5% means an unobserved confounder would need to explain at least 17.5% of the residual variation in both FL presence and log prices to reduce the coefficient to zero.

Robust to omitted variable bias

The robustness value \(RV_{q=1} = 17.5\%\) is a stringent requirement given the rich fixed effects structure (item + year + PBU).

Oster Coefficient Stability¶

The Oster (2019) bound is degenerate (\(\hat{\delta} = 261.6\)) because PBU FE barely move \(R^2\) (0.879 to 0.886)---a design strength, not fragility. The Cinelli--Hazlett framework (\(RV = 17.5\%\)) is the appropriate sensitivity metric.

4. Staggered Difference-in-Differences¶

The staggered DiD is reported as a complementary exercise. Positive pre-trends at \(t = -2\) and \(t = -3\) preclude a causal reading, consistent with strategic market selection.

Callaway & Sant'Anna¶

Metric	Value
ATT (log prices)	0.014 (SE = 0.039)
Status	Positive but insignificant

Stacked DiD¶

Metric	Value
Coefficient	\(-0.006\) (SE = 0.014)
\(N\)	715,116
Status	Pre-trends preclude causal reading

Non-causal framing

The pre-trends are consistent with strategic market selection but equally compatible with unobserved confounding. The paper relies on within-item conditional comparisons for this reason, not on the DiD.

Event study — **Figure 10.** TWFE event study: coefficients on years relative to first FL entry, by outcome variable.

5. IV as Measurement-Error Diagnostic¶

The leave-one-out IV is reported as a measurement-error diagnostic, not a preferred estimate. Exclusion-restriction concerns keep it off the primary 3.6--7.7% range.

DV	OLS	IV	First-stage \(F\)
Log price	0.064	0.194	396
Log firms	0.167	0.501	396
Log bids	0.169	0.614	396
Log non-FL firms	0.143	0.404	396

The implied signal-to-total-variance ratio:

\[\hat{\lambda} = \frac{\hat{\beta}_{\text{OLS}}}{\hat{\beta}_{\text{IV}}} = \frac{0.064}{0.194} \approx 0.33\]

Measurement-error interpretation

Given that the FL dummy is a deliberately coarse screen defined by a distributional threshold, a signal-to-noise ratio of one-third is plausible. The OLS estimate captures the conditional association; the IV points to attenuation in the binary indicator. Neither is causal.

Balance Tests¶

Pre-determined observables regressed on the LOO instrument with item, year, and PBU FE yield standardized differences below 0.03\(\sigma\)---statistically significant but economically negligible (\(< 0.001\)).

6. Oversight Heterogeneity¶

The FL--price association varies sharply with PBU size: a 12.5x gradient across quartiles.

PBU Size Quartile	Coefficient	Interpretation
Q1 (smallest)	0.214	Weakest oversight
Q2	0.098
Q3	0.045
Q4 (largest)	0.017	Strongest oversight

The monotonic decline matches the framework's comparative static (\(\partial m^*/\partial \theta_k < 0\)): where oversight is weaker, the screen bites harder. Smaller PBUs also have thinner fixed-effect cells, so part of the gradient could be mechanical, but the monotonic decline across all four quartiles is hard to attribute to noise alone.

Oversight heterogeneity — **Figure 11.** FL price coefficient by oversight proxy (PBU size quartile and procedure type).

Robustness Summary Table¶

Check	Coeff.	SE	\(N\)	Assumption
OLS (item + year + PBU FE)	0.064	0.020	1,654,401	Sel. on obs. + FE
CEM	0.077	0.024	969,751	Sel. on obs.
IPW	0.055	0.021	830,194	Sel. on obs.
Cross-fit	0.036	0.019	1,654,401	Sel. on obs. + FE; no mechanical link
IQR 1.0x	0.079	0.023	1,654,401	Sel. on obs. + FE
IQR 2.0x	0.060	0.022	1,654,401	Sel. on obs. + FE
IQR 3.0x	0.050	0.024	1,654,401	Sel. on obs. + FE
Item x year FE	0.074	0.022	1,654,401	Sel. on obs. + FE
Two-way clustering	0.064	0.024	1,654,401	Sel. on obs. + FE
Horse-race (FL + CV)	0.084	0.023	1,654,401	Sel. on obs. + FE
Continuous (log max AL)	0.022	0.005	1,654,401	Sel. on obs. + FE
IV (leave-one-out)	0.194	0.077	1,654,401	Excl. restriction
Stacked DiD	\(-0.006\)	0.014	715,116	Parallel trends
Oster \(\hat{\delta}\)	degen.	--	1,654,401	\(R^2\) gap \(\approx\) 0