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.

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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.

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