H:parallel-trends-hold — Reduced-form DiD validates timing and sign; clean parallel trends are NOT claimed, and identification is carried by the structural decomposition¶
The DiDiR design compares the switched group 65 to the always-treated control groups. A strict reduced-form-only reading would require that, absent the policy reversal, group 65 prices and participation would have moved in parallel with the controls. The paper does not claim that. The joint pre-trend test rejects flatness for both participation margins, and Group 65 is not balanced enough against the pooled controls to support a reduced-form-only design. The reduced form is therefore demoted to a timing, sign, and approximate-scale check; the identification work is carried by the structural price-formation decomposition. The participation event study is reported as Online Appendix Figure OA-1.
Intuition (plain-language)
The design would like Group 65 prices to have tracked the control groups before the reform, but they did not cleanly: the joint pre-trend test rejects flatness on both participation margins (partly because the platform-enablement months in late 2017 fall inside the pre-window), and pre-period covariates are imbalanced. So the paper does not lean on clean parallel trends. Instead, the reduced-form DiD is used only to pin down timing, sign, and rough scale — non-SME participation drops sharply right at the cutoff and prices move in the predicted direction — while the actual magnitude and welfare claims come from the structural decomposition. A reassuring consistency check: the structural net effect (0.227 of the reference price) scaled by the 43% realized SME-only adherence lands near 10%, the same order of magnitude as the 10–11% reduced-form DiD.
Evidence strength: Partial — timing and sign supported; clean parallel trends NOT claimed. The reduced form validates the direction and timing of the effect, not a clean-trend causal magnitude. The joint pre-trend test rejects flatness for both participation margins (SME \(F = 19.08\), non-SME \(F = 38.97\), both \(p = 0.000\)), partly reflecting the BEC platform-enablement months in late 2017 that fall inside the pre-window; the OA-1 event study is read for timing and direction, not as a clean-trend estimate (AN-023). Group 65 is also imbalanced on pre-period covariates, which is why the reduced form is not asked to carry the structural magnitude or the welfare ranking. What the reduced form does establish converges across estimators: Sun-Abraham (item-level) ATT 0.108 reproduces DDR 0.113 within 0.005 on log prices (AN-018); Callaway-Sant'Anna at group×month gives 0.237 — same direction, wider SE; BJS imputation attenuates but keeps the sign; modern-DiD attenuation is itself expected under heterogeneous timing. The Goodman-Bacon decomposition reports 100% weight on the clean Treated-vs-Untreated 2×2 (AN-020), and placebo cutoffs are smaller in magnitude than the real cutoff (AN-004). Lee bounds (AN-005) and HonestDiD (AN-006) further discipline the reduced-form coefficient. The bottom line: the DiD corroborates the sign and timing of the predicted price footprint, but identification of the mechanism and its magnitude is structural, not reduced-form.
Theory¶
A reduced-form-only reading of the DiDiR coefficient would require parallel trends in the absence of the policy reversal. The paper does not rely on that. The pre-trend battery is run to characterize the reduced form, not to certify a clean-trend causal estimate: - Joint pre-trend test: the event-study pre-period coefficients should be jointly flat if parallel trends held. They are not — the test rejects flatness on both participation margins, so the reduced form is used only for timing/sign/scale. - Pre-treatment placebos: applying the DiDiR specification to a fake pre-treatment date should yield coefficients smaller in magnitude than the real cutoff. - HonestDiD: Rambachan and Roth (2023) robust CIs under bounded post-treatment trend deviation M̄. - Lee bounds: Lee (2009) bounds correct the conditional price coefficient against differential completion.
Because clean parallel trends are not assumed, identification of the mechanism and its magnitude is delegated to the structural price-formation decomposition (see H:exclusion-dominates). A structural↔DiD consistency check ties them together: the structural net effect of 0.227 of the reference price, scaled by the 43% realized SME-only adherence, is ≈ 10% — the same order of magnitude as the 10–11% reduced-form DiD.
Prediction¶
(i) The joint pre-trend test will reject flatness — the reduced form is not a clean-trend design and is therefore restricted to timing/sign/scale. (ii) The non-SME participation event study (Online Appendix Fig. OA-1) will show a sharp drop at the cutoff, supporting the timing/direction reading. (iii) Placebo cutoffs will be smaller in magnitude than the real cutoff; modern-DiD estimators (SA, CS, BJS) will preserve the sign while attenuating the magnitude. (iv) Lee bounds and HonestDiD will keep the reduced-form price coefficient disciplined.
Competing prediction¶
A clean-trend causal magnitude. A reviewer might expect the page to claim that flat pre-trends license reading the DiDiR coefficient as the causal magnitude. The paper deliberately does not make that claim: the joint pre-trend test rejects flatness (partly because the late-2017 platform-enablement months fall inside the pre-window), and pre-period covariates are imbalanced. The reduced form is therefore demoted to a timing/sign/scale check, and the magnitude comes from the structure. The firm-count and bid-count pre-trends also partly reflect the control groups' gradual SME-restriction rollout, not a group-65-specific shock — see H:entry-thickens-pool.
Setting evidence¶
The pre-treatment period is long enough to test trend stability. The placebo design uses Sep 2017, Mar 2017, and Jun 2017 as fake treatment dates, splitting the pre-period into nominal "pre" and "post" halves while applying the same DiDiR specification.
Empirical test¶
- Joint pre-trend test (participation event study, Online Appendix Fig. OA-1): an F-test on the pre-period event-study coefficients. It rejects flatness on both margins (SME F=19.08, non-SME F=38.97, both p=0.000), so the reduced form is read for timing and direction only.
- Pre-treatment placebos: DiDiR with fake treatment dates Sep 2017, Mar 2017, and Jun 2017; price coefficient should be small relative to the real cutoff.
- HonestDiD: Rambachan-Roth bounds on the post-treatment trend deviation; the price effect should remain significant for moderate M̄.
- Lee bounds: Lee (2009) selection-bound formula applied to the conditional price coefficient under differential completion rates.
- Staggered-DiD estimators: Callaway-Sant'Anna (2021) and Sun-Abraham (2021) reproductions to address heterogeneous-timing bias.
Data requirements and limitations¶
The placebo test uses pre-treatment data only; the post-treatment period is excluded from the test by construction. HonestDiD requires the event-study specification (already in docs/results.md). Lee bounds require the completion indicator (already in BEC). Callaway-Sant'Anna requires defining treatment cohorts; since group 65 is a single switching event, the staggered application is to the control groups' rollout history, not to group 65 itself.
Evidence¶
| Analysis | Bearing | Key takeaway |
|---|---|---|
| AN-023 | Qualifies | Joint pre-trend test rejects flatness on both participation margins (SME F=19.08, non-SME F=38.97, both p=0.000), partly reflecting late-2017 platform-enablement months in the pre-window. OA-1 event study read for timing/direction, not clean trends. Dropping the 8-month enablement window moves β from −0.109 to −0.087 (−22%); both remain p<0.01 — timing/sign robust, not a clean-trend magnitude. |
| AN-004 | Supports | Placebo cutoffs (Sep / Mar / Jun 2017) smaller in magnitude than the real cutoff — consistent with a timing/sign reading of the reduced form. |
| AN-005 | Supports | Bounds −0.131 to −0.123, both significant; differential completion has negligible impact on the conditional coefficient. |
| AN-006 | Supports | Reduced-form price coefficient remains disciplined under M̄ violations. |
| AN-018 | Supports | Sun-Abraham item-level ATT 0.108 vs DDR 0.113 (Δ<0.005 on log prices); CS2021 group-month 0.237 same direction; BJS attenuates but keeps sign — sign/timing convergence, not a magnitude claim. |
| AN-020 | Supports | Weight = 1.000 on clean Treated-vs-Untreated 2×2; zero forbidden comparisons. Heterogeneous-timing bias mathematically absent in this single-cohort design. |
| AN-021 | Mixed | Ridge-augmented synthetic control matches pre-period by construction; post ATT 0.171 directionally consistent; placebo p=0.103 borderline. Directional corroboration only — does not certify clean trends. |
Open tests¶
Randomization inference at the group level¶
scripts/05_robustness.R runs a 500-iteration permutation test
reassigning the group 65 indicator across groups. The figure already
exists (docs/robustness.md). Worth promoting to a
separate AN page covering the entire RI battery.
Tighten the synthetic-control placebo \(p\)¶
AN-021 reports placebo \(p = 0.103\) (borderline). Two follow-ups would tighten it: (i) augmented synthetic control with covariates (item-mix, PBU composition); (ii) quarterly rather than semester aggregation (12 vs 6 pre-period observations). Either would push the placebo \(p\) toward conventional significance.