Advanced Methods¶
This page documents the dynamic and identification machinery that supports the analysis. These methods are reported as diagnostics: they help with timing and sensitivity, but they are not the primary identification. The main claims rest on the Lee selection bounds, the within firm-buyer-item pricing test, and the sourcing evidence.
1. Dynamic event study (BJS) and Honest-DiD sensitivity¶
The dynamic evidence is useful for timing, not for the primary identification strategy. The BJS event-study estimator (Borusyak, Jaravel, and Spiess, 2024) traces the litigated-procurement price path around exposure; Honest-DiD (Rambachan and Roth, 2023) tests how robust the early estimate is to controlled deviations from parallel trends.
| Estimate | Coefficient | SE |
|---|---|---|
| BJS, first post-exposure period (t0) | 0.052 | (0.018) |
| BJS, five periods after exposure (t5) | 0.147 | (0.026) |
The first post-period estimate does not survive Honest-DiD deviations at the observed maximum pre-period scale. Because the dynamic design is sensitive to plausible deviations from parallel trends, it is used as diagnostic evidence rather than as a basis for the main claims.
Economic intuition
A rising event-study path is seductive but easy to over-read: it identifies a causal dynamic only if pre-trends would have stayed flat absent treatment. Honest-DiD asks how large a violation of that assumption it would take to overturn the early estimate — and here a violation no bigger than the observed pre-period scale suffices. So the dynamics are kept for timing, while identification rests on the bounds, the within-firm test, and the sourcing evidence instead.
Detail: AN-010 — Dynamic BJS and Honest-DiD.
2. Lee trimming bounds¶
The price comparison conditions on a screened, overrepresented administrative channel. Lee (2009) trimming bounds discipline that selection within item-by-year-by-PBU strata, trimming the high and low tails of the administrative price distribution to produce lower and upper bounds for the litigated-over-administrative gap. The bounds discipline selection under a monotonicity restriction; they do not eliminate it or point-identify a counterfactual with sanctions removed. The selection-corrected interval is [15.9%, 21.1%].
Economic intuition
When the comparison group is selected, a point estimate is a guess dressed as a fact. Lee bounds give up point identification on purpose: trimming the tails of the over-represented administrative group brackets the true gap under a single monotonicity restriction. Honest partial identification beats a fragile point estimate — the interval is the credible object.
Detail: AN-002 — Lee bounds.
3. Wild-cluster bootstrap¶
With few and uneven PBU clusters, conventional cluster-robust inference can be unreliable. Rademacher wild-cluster bootstrap inference re-tests the under-the-gun contrast: the preferred specification gives p = 0.0080, and the tighter item-by-year-month specification gives p = 0.0390. The bootstrap addresses few-cluster inference; it does not remove the selection concern handled by the bounds.
Economic intuition
Cluster-robust asymptotics assume many clusters; with few, the usual reference distribution is wrong and tests over-reject. The wild bootstrap rebuilds the null distribution by reweighting cluster residuals, so inference no longer leans on a large cluster count. It buys credible p-values — and says nothing about bias or selection, which are handled elsewhere.
Detail: AN-007 — Wild-cluster bootstrap.
These three methods support the analysis but are not its primary identification. They establish timing (BJS, as a diagnostic), discipline selection (Lee bounds), and stabilize inference (wild-cluster bootstrap) around claims that the main design already carries.