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Diferences in Diferences Model

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Difference in Differences Model to estimate treatment effects
Mayank Nagpal
This week’s readings discuss the Difference in Differences model which is used to test the and estimate the effect of a particular treatment applied to the treatment group by looking at how the change in trend before and after the application of the treatment. To estimate the effect of a treatment we aim to estimate the difference between a person in a world in which the treatment was applied and a world where the treatment is not applied. In the real world however, only one is observable. Thus, we compare two different groups of individuals or subject where the pre-treatment trends are similar. In other words, the Difference in Differences model compares the change in trend in the control group and the treatment group given the assumption that the pretreatment trend in the two groups are similar. The basic assumption in the model is that if the treatment had not been applied, the control and the treatment group would show similar trends in the post-treatment time period. Thus, any difference in seen in the trend for treatment and control groups can be attributed to the treatment being applied to the treatment group.
A fixed effects model which uses a dummy for the treatment vs control group and fixed effects for time and individuals is inefficient for the estimating the effect of the treatment as the individual differences may be misreported and be just noise. This may result in a measurement error caused by considering individual effects in the model. A differences in differences model thus uses fixed effects for the treatment and control group instead of the individual level fixed effects.
The model looks for people with the same pre-treatment trends in the outcome variable and uses them as the control group. Suppose, we have two periods, pre and post treatment representing time, t (t=post, pre) and the group (G) which an individual belongs to (G=treat, control). Then, under the assumption that the trends in the treatment and control groups would have continued the same way as before in the absence of treatment, we estimate the treatment effect as
Treatment Effect=[EYigtt=post, G=treat, Xi)- EYigtt=pre, G=treat, Xi)] - [EYigtt=post, G=control, Xi)- EYigtt=pre, G=control, Xi)]
Where, Xi is a vector of individual level control covariates and E(Yigt|t,G,Xi) is the expected value of the outcome given group (G), time period (t) and the values of Xi.
The effect of the treatment can be estimated using the below regression:
Yit=β1+β2*Treati+β2*Postt+ ρ*(Treat*Post)it+δ*Xit+εit
Where, Treat represents the dummy for the treatment group and Post is the dummy for the post-treatment time period. ρ, the coefficient for the interaction term for the two dummies provides the estimate of the treatment effect on the outcome. Xit represents the control covariates for individual i at time t

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