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Suppose that Walmart has 1,000 stores. It has a 20% coupon for cereal, and it hypothesizes that the coupon will increase the sales of cereal by 3%.

Walmart put the coupon in 100 stores on 2022-05-01; the other 900 stores continue to have no coupon. Unfortunately, it does NOT have any sales data from before 2022-05-01. The only data that it has is in the post-intervention period (from 2022-05-01 till today).

Assume that I have data on all the confounding variables that you care about.

I was going to use difference-in-difference to estimate the average treatment effect, but I have no pre-intervention data.

What I can do instead to estimate the average treatment effect?

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    $\begingroup$ All you have is data from when the coupon is active. There is nothing to compare it to. For difference in differences you need a comparison group and a pre-treatment group. So you can't do that with the data you have. $\endgroup$
    – Noah
    Commented May 7, 2023 at 21:08
  • $\begingroup$ The other 900 stores have no coupon, so there is a control group. There just isn't a pre-treatment period. Is it possible to still estimate effect of the coupon? $\endgroup$
    – Iterator516
    Commented May 7, 2023 at 21:37
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    $\begingroup$ @Iterator516 So long as the stores are randomly selected and you are willing to make very very strong assumptions about exchangeability of stores and their counterfactual outcomes, you can compare the post outcomes directly. However, given what we know about economics and differences in geography I'm willing to bet these assumptions are probably not worth making. $\endgroup$
    – Demetri Pananos
    Commented May 8, 2023 at 5:12
  • $\begingroup$ They are not randomly selected. You are correct, the exchangeability assumption is not valid in this scenario, because there are too many heterogeneous factors affecting the stores (climate, driving distance, walking distance, store manager, neighborhood income, store size, store design, shopper demographics, just to name A FEW). $\endgroup$
    – Iterator516
    Commented May 11, 2023 at 1:10

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