When you want to measure the causal impact of an intervention, there are many different ways. Differences of Difference (DiD) and Comprehensive Control Method (SCM) are two commonly used methods, and I have provided a summary of the econometric overview of each method (Have done, Single chip microcomputer).a dissertation Arkhangelski et al (2021) exist Air Summarize the differences

Generally speaking, the DID method is suitable for situations where we have a large number of units exposed to policy. Researchers are willing to make a “parallel trend” hypothesis, which means that we can fully control the unit-specific and time-specific selection effects by considering additive factors. Fixed effects. In contrast, the SC method introduced in environments where only one (or a small number) of units are exposed attempts to compensate for the lack of parallel trends by reweighting the units to match their pre-exposure trends.

Arkhangelsky and co-authors also believe that the answer should not be whether to use DiD or MCU; but we should use DiD and SCM uses their proposed synthetic differential difference (SDiD) estimator. When there are binary exposure variables in which the control unit never accepts intervention and the treatment unit has a pre-exposure and post-exposure period, the SDiD parameter can be estimated as follows.

In this equation, the result is Y_it, Term ? ?_{A generation}, ?_Ton They are intercept, individual fixed effects, and time trends.The key parameter of interest is ?, which interacts with the indicator variable to determine whether the individual A generation In time A generation In the treatment group. The method in the innermost bracket is similar to the difference in difference estimation. The key difference is that the difference in this difference is weighted by two terms: ?_{A generation}^standard And ?_Ton^standard Is the weight, where ?_{A generation}^standard Align the pre-exposure trend of the results of the unexposed units with the results of the exposed units and ?_Ton^standard Balance the time period before exposure and the time period after exposure. This method is similar to the SCM estimator, but has the individual fixed effect of SCM, ?_{A generation}, Is omitted and has a weight ?_{A generation}^{Supply Chain Management}, Ie use.

The main advantage of this method is that it can better estimate causal effects, because the weights make the comparison more “local”, because the method gives more weight pairs to units that are similar to the processed units and put options in the past and the processing period is similar The period has more weight. This makes the estimation more robust and also improves the accuracy of the estimation. The disadvantage of this method is that the estimation is “partial” and the results should be extrapolated carefully to a complete unweighted sample.sample

This full text Describes in more detail how to estimate unit weights, time weights, and regularization parameters. Specifically, the unit weight ?_{A generation}^standard Calculations are made so that the weighted control group results during the pre-exposure period have similar trends to the observed unweighted pre-exposure treatment group results. The regularization parameter is designed to match the magnitude of the typical single-period change of unexposed units in the previous period, multiplied by the theoretically motivated scale (N_trTon_postal)^1/4. The regularization parameter is used to increase the dispersion when calculating the unit weight. In order to estimate time weights, the sample is restricted to the observations of the control group and the time period of the control group results before treatment is weighted, so they look similar to the results of the control group after treatment.