31.2 Estimating QTT with CiC

CiC relies on four distributions from a 2 × 2 Difference-in-Differences (DiD) setup:

  1. \(F_{Y(0),00}\): CDF of \(Y(0)\) for control units in period 0.
  2. \(F_{Y(0),10}\): CDF of \(Y(0)\) for treatment units in period 0.
  3. \(F_{Y(0),01}\): CDF of \(Y(0)\) for control units in period 1.
  4. \(F_{Y(1),11}\): CDF of \(Y(1)\) for treatment units in period 1.

The Quantile Treatment Effect on the Treated (QTT) at quantile \(\theta\) is:

\[ \Delta_\theta^{QTT} = F_{Y(1), 11}^{-1} (\theta) - F_{Y (0), 11}^{-1} (\theta) \]

To estimate the counterfactual CDF:

\[ \hat{F}_{Y(0),11}(y) = F_{y,01}\left(F^{-1}_{y,00}\left(F_{y,10}(y)\right)\right) \]

This leads to the estimation of the inverse counterfactual CDF:

\[ \hat{F}^{-1}_{Y(0),11}(\theta) = F^{-1}_{y,01}\left(F_{y,00}\left(F^{-1}_{y,10}(\theta)\right)\right) \]

Finally, the treatment effect estimate is:

\[ \hat{\Delta}^{CIC}_{\theta} = F^{-1}_{Y(1),11}(\theta) - \hat{F}^{-1}_{Y(0),11}(\theta) \]

Alternatively, CiC can be expressed as the difference between two QTE estimates:

\[ \Delta^{CIC}_{\theta} = \Delta^{QTE}_{\theta,1} - \Delta^{QTE}_{\theta',0} \]

where:

  • \(\Delta^{QTT}_{\theta,1}\) represents the change over time at quantile \(\theta\) for the treated group (\(D=1\)).
  • \(\Delta^{QTU}_{\theta',0}\) represents the change over time at quantile \(\theta'\) for the control group (\(D=0\)).
    • The quantile \(\theta'\) is selected to match the value of \(y\) at quantile \(\theta\) in the treated group’s period 0 distribution.

Marketing Example

Suppose a company introduces a new online marketing strategy aimed at improving customer retention rates. The goal is to analyze how this strategy affects retention at different quantiles of the customer base.

  1. QTT Interpretation:
    • Instead of looking at the average effect of the marketing strategy, CiC allows the company to examine how retention rates change across different quantiles (e.g., low vs. high-retention customers).
  2. Rank Preservation Assumption:
    • This approach assumes that customers’ rank in the retention distribution remains unchanged, regardless of whether they received the new strategy.
  3. Counterfactual Distribution:
    • CiC helps estimate how retention rates would have evolved without the new strategy, by comparing trends in the control group.