AN-005: Lee bounds on the DiDiR price coefficient¶
Intuition (plain-language)
Different auctions complete under the two regimes, so maybe selection — not price formation — drives the result. Lee bounds draw a worst-case box around that concern. The box is tiny here (−0.131 to −0.123), so differential completion cannot explain away the price effect.
Reduced-form motivation layer
The numbers below are from the v1–v4 reduced-form DiDiR pipeline
(scripts/02_analysis.R + companions), which the v8 manuscript
carries as motivation in §1 but does not headline. The canonical
v8 result is the structural counterfactual decomposition — see
AN-010 (decomposition) and
AN-011 (welfare arithmetic).
Question¶
The price and distance regressions condition on completion (i.e., the item was successfully procured). If the treatment shifts completion rates, the conditional coefficient confounds the policy effect with differential selection into the sample. Lee (2009) bounds correct for this by trimming the outcome distribution in the excess-selected cell.
Design¶
- Sample: 18-month window; both completed and non-completed items
(completion indicator from
oc_item_status). - Variation: same DiDiR as AN-001.
- Specification: Lee (2009) trimming bounds — the conditional outcome distribution in the cell with the higher completion rate is trimmed at the proportion required to match the lower-completion cell, producing tight bounds on the conditional coefficient.
- Outcomes: log prices, distance.
Results¶
| Outcome | Lower bound | Upper bound | Trimming proportion |
|---|---|---|---|
| Log prices | −0.1306*** (0.0096) | −0.1227*** (0.0085) | (see note) |
| Distance (km) | +10.775*** (2.232) | +10.804*** (2.233) | 8.82% |
Item-clustered SE. *** p<0.01.
Output: output/tables/tab_lee_bounds.tex.
Interpretation¶
The bounds on log prices are tight: a gap of ~0.008 between lower and upper endpoints, well below the standard error of the point estimate. Both endpoints reject zero at p<0.01. Sample selection through differential completion has negligible impact on the price coefficient.
The distance bounds are essentially identical at both endpoints (+10.78 vs +10.80 km), with an 8.82% trimming proportion. The distance estimate is therefore not driven by selection on completion.
Confidence: yellow. The Lee bounds discipline the selection-on-completion margin; they do not address other identification threats (parallel trends, cost shocks). The full identification battery is in AN-004, AN-005, AN-006. The reading is yellow because all three are own-project robustness checks rather than independent replications.
Follow-ups¶
- Composition-margin Lee bound: extend the trimming logic to the SME-winner indicator (AN-009) to bound the composition-shift coefficient against differential completion.
- Two-margin selection: completion and item-attribute coverage may both be selective. A joint selection model (or a Heckman two-step) would extend the Lee bound; not in scope for the current paper.