How do you estimate causal effects using difference-in-differences?

1 Assumptions of DID

To use DID, you need to make some assumptions about your data and your treatment variable. The most important one is the parallel trends assumption, which means that in the absence of the treatment, the two groups would have followed the same trend over time. This implies that the treatment is the only factor that causes a divergence in the outcomes between the groups. You can check this assumption by plotting the pre-treatment trends of the groups and looking for any systematic differences. Another assumption is that the treatment is stable and does not vary over time or across units. This means that you can ignore any other factors that might affect the outcomes besides the treatment.

2 Formula of DID

The basic formula of DID is: DID = (Y1t - Y0t) - (Y1c - Y0c) where Y is the outcome variable, t and c are the treatment and control groups, and 1 and 0 are the post-treatment and pre-treatment periods. This formula calculates the difference in the average outcomes between the two groups after the treatment, and subtracts the difference in the average outcomes between the two groups before the treatment. The result is the estimated causal effect of the treatment on the outcome. You can also express this formula as a regression model, where you include dummy variables for the group, the time period, and their interaction, and control for other covariates if needed.

3 Example of DID

To illustrate how DID works, let's use a simple example. Suppose you want to estimate the effect of a new law that bans smoking in public places on the smoking rate of adults. You have data on two states, A and B, where state A implemented the law in 2019, and state B did not. You have the smoking rate of adults in both states for 2018 and 2020. You can use DID to compare the changes in smoking rate between the two states over time. The DID formula is: DID = (Y1A - Y0A) - (Y1B - Y0B) where Y is the smoking rate, A and B are the states, and 1 and 0 are the years 2020 and 2018. Suppose the data are: Y0A = 20%, Y1A = 15%, Y0B = 25%, Y1B = 24% Then, the DID estimate is: DID = (15% - 20%) - (24% - 25%) = -5% - (-1%) = -4% This means that the new law reduced the smoking rate by 4 percentage points in state A compared to state B. You can interpret this as the causal effect of the law on the smoking rate, assuming that the parallel trends and other assumptions hold.

4 Advantages and limitations of DID

Difference-in-differences (DID) has advantages and limitations as a causal inference method. It is easy to implement and understand, as it only requires data on two groups and two time periods, and does not necessitate strong assumptions about the functional form of the relationship between the outcome and the treatment. Furthermore, DID can control for any confounding factors that are fixed over time and across groups. However, it requires the parallel trends assumption, which might not hold in some cases. Also, DID can be sensitive to outliers, measurement errors, or missing data, as they can affect the differences in means. Finally, it can be biased by spillover effects or dynamic effects.

5 Extensions and variations of DID

DID can be extended and modified to address some of the limitations and challenges of the basic method. For example, synthetic control creates a weighted combination of control units that mimics the pre-treatment trend of the treated unit, while generalized DID uses more than two groups and two time periods. Additionally, an event study examines the effects of the treatment before and after the treatment period to test for anticipatory or delayed responses. Finally, regression discontinuity exploits a discontinuous change in the treatment assignment based on a threshold or a cutoff value of a variable.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?