DID_HAD: Stata module to estimate the effect of a treatment on an outcome in a heterogeneous adoption design with no stayers but some quasi stayers

did_had estimates the effect of a treatment on an outcome in a heterogeneous adoption design (HAD) with no stayers but some quasi stayers. HADs are designs where all groups are untreated in the first period, and then some groups receive a strictly positive treatment dose at a period F, which has to be the same for all treated groups (with variation in treatment timing, the did_multiplegt_dyn package may be used). Therefore, there is variation in treatment intensity, but no variation in treatment timing. HADs without stayers are designs where all groups receive a strictly positive treatment dose at period F: no group remains untreated. Then, one cannot use untreated units to recover the counterfactual outcome evolution that treated groups would have experienced from before to after F, without treatment. To circumvent this, did_had implements the estimator from de Chaisemartin and D'Haultfoeuille (2024) which uses so-called "quasi stayers" as the control group. Quasi stayers are groups that receive a "small enough" treatment dose at F to be regarded as "as good as untreated". Therefore, did_had can only be used if there are groups with a treatment dose "close to zero". Formally, the density of groups' period-two treatment dose needs to be strictly positive at zero, something that can be assessed by plotting a kernel density estimate of that density. The command makes use of the lprobust command by Calonico, Cattaneo and Farrell (2019) to determine an optimal bandwidth, i.e. a treatment dose below which groups can be considered as quasi stayers. To estimate the treatment's effect, the command starts by computing the difference between the change in outcome of all groups and the intercept in a local linear regression of the outcome change on the treatment dose among quasi-stayers. Then, that difference is scaled by groups' average treatment dose at period two. Standard errors and confidence intervals are also computed leveraging lprobust. We recommend that users of did_had cite de Chaisemartin and D'Haultfoeuille (2024), Calonico, Cattaneo and Farrell (2019), and Calonico, Cattaneo and Farrell (2018).