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Advanced Methods

This page documents five advanced econometric methods that complement the main DiDiR analysis. Each method addresses a specific concern or deepens the understanding of the treatment effect.

The HonestDiD method (Rambachan and Roth, 2023) constructs robust confidence intervals for the treatment effect under controlled violations of the parallel trends assumption. The parameter \(\bar{M}\) governs the maximum amount by which the post-treatment trend may deviate from the pre-treatment path.

HonestDiD sensitivity analysis
Figure A.10. HonestDiD Sensitivity Analysis: Robust confidence intervals under violations of parallel trends. The price effect remains significant even for large values of M̄.

Key finding

The main price effect remains statistically significant even under substantial deviations from strict parallel trends, providing strong evidence that the results are not an artifact of pre-existing differential trends.

2. Sample Selection Correction (Lee Bounds)

The price and distance regressions condition on item completion. If the treatment affects completion rates, this creates sample selection bias. Lee (2009) bounds correct for this by trimming the outcome distribution in the excess-selected cell.

Log prices (lower) Log prices (upper) Distance (lower) Distance (upper)
g65 x Pre -0.1306*** -0.1227*** 10.775*** 10.804***
(0.0096) (0.0085) (2.232) (2.233)
Trimming proportion 8.82%

Key finding

The bounds for log prices range from -0.131 to -0.123, both highly significant. The tightness of the bounds indicates that sample selection has a negligible impact on the main price estimates.

3. Heterogeneous Treatment Effects (Causal Forest)

A causal forest (Athey, Tibshirani, and Wager, 2019) estimates individualized treatment effects using an honest, doubly-robust random forest on FWL-residualized outcomes.

Variable Importance

Causal forest variable importance
Figure A.11. Variable importance ranking: item quantity dominates (0.486), followed by tender type (0.183) and supplier location (0.156).

Group Average Treatment Effects (GATE)

Causal forest GATE
Figure A.12. Group Average Treatment Effects by CATE quartile. The heterogeneity across quartiles is modest.
Q1 (lowest) Q2 Q3 Q4 (highest)
GATE 0.391 -0.114 1.048 -0.824
(0.941) (1.363) (1.374) (0.844)
ATE (full sample) 0.125 (0.578)

Interpretation

The causal forest reveals modest heterogeneity in treatment effects, with item quantity as the key moderator. The imprecise GATE estimates reflect the challenge of detecting heterogeneity in residualized data with substantial within-item variation.

4. Distributional Effects (Quantile DiD)

Quantile difference-in-differences (Canay, 2011) estimates how the treatment effect varies across the price distribution, going beyond the mean effect captured by OLS.

Quantile DiD
Figure A.13. Quantile DiD coefficients across the price distribution. Effects are strongly negative at lower quantiles and turn positive at the upper tail.
\(\tau = 0.10\) \(\tau = 0.25\) \(\tau = 0.50\) \(\tau = 0.75\) \(\tau = 0.90\)
g65 x Pre -0.623*** -0.543*** -0.356*** 0.031 0.445***
(0.028) (0.023) (0.024) (0.032) (0.045)
OLS benchmark -0.124 (0.017)

Key finding

The benefits of open competition are concentrated at the lower quantiles of the price distribution (\(\tau \leq 0.50\)), where competitive bidding drives prices down most effectively. At the upper tail (\(\tau = 0.90\)), prices are actually higher under open tenders, possibly reflecting specialized items with thin supplier markets.

5. Mechanism Decomposition (Gelbach)

The Gelbach (2016) decomposition partitions the total price effect into contributions from observable channels by comparing a "short" regression (treatment + controls) with a "full" regression that adds mediators.

Gelbach decomposition
Figure A.14. Channel contributions to the price effect. Competition and composition operate as partially offsetting mechanisms.
Channel Coefficient SE % of gap
Short regression (g65 x Pre) -0.1318*** (0.0096)
Full regression (g65 x Pre) -0.1227*** (0.0101)
Gap (short - full) -0.0091 100%
Competition (log firms) 0.0078*** (0.0010) -85%
Composition (SME winner) -0.0169*** (0.0014) 185%

Interpretation

The competition and composition channels operate as partially offsetting mechanisms. Open tenders increase competition (lowering prices), but also attract non-SME winners whose conditional pricing effects partially offset the competition gains. The large unexplained component (-0.123) indicates that most of the price effect operates through channels not captured by these two mediators.