Question

Explain how you can use difference in difference to identify causal identification in a research paper

Answer 1

Here, we use g = 1 ... G to index cross-sectional units and t = 1 ...T to index time periods. In DID studies, g often refers to geographical areas such as states, counties or census tracts, although it could also refer to distinct groups such as those separated by age Most of the time, t represents years, quarters, or months. In most applications, researchers are concerned
with outcomes in two alternative treatment regimes: the treatment condition and the control
condition. To make the idea concrete, let Dgt = 1 if unit g is exposed to treatment in period t,
and Dgt = 0 if unit g is exposed to the control condition in period t. In public health applications, the set of treatments might consist, for example, of two alternative approaches to the regulation of syringe exchange programs that are adopted in different states in different years .
Research on the causal effects of the treatment condition revolves around the outcomes that
would prevail in each unit and time period under the alternative levels of treatment. One way to
make this idea more tangible is to define potential outcomes that describe the same unit under different (hypothetical) treatment situations. To that end, let Y(1)gt represent an outcome of interest for unit g in period t under a hypothetical scenario in which the treatment was active in g at t; Y(0)gt is the outcome of the same unit and time under the alternative scenario in which the control condition was active in g at t. The treatment effect for this specific unit and time period is -gt = Y(1)gt − Y(0)gt, which is simply the difference in the value of the outcome variable for the same unit across the two hypothetical situations. The notation suggests this would be easily done, but applied researchers cannot observe the identical unit under two different scenarios as one
could through a lab experiment; in practice, each unit is exposed to only one treatment condition
in a specific time period, and we observe the corresponding outcome. Specifically, for a given unit and time, we observe Ygt = Y(0)gt + [Y(1)gt − Y(0)gt]Dgt. The notation so far describes the counterfactual inference problem that arises in every causal inference study. In a typical study, researchers have access to data on Ygt and Dgt, and they aim to combine the data with research design assumptions to learn about the average value of Y(1)gt − Y(0)gt in a study population. The DID design is a quasi-experimental alternative to the well-understood and straightforward RCT design, seen for example in the health insurance context in
the RAND Health Insurance Experiment in the 1970s and more recently in the Oregon Health
Insurance Experiment RCT and DID share some characteristics: Both involve a well-defined study population and set of treatment conditions, where it is easy to distinguish between a treatment group and a control group and between pretreatment and post-treatment time periods. The most important distinction is that treatment conditions are randomly assigned across units in an RCT but not in a DID design. Under random assignment, treatment exposure is statistically independent of any (measured or unmeasured) factor that might also affect outcomes. In a DID design, researchers
cannot rely on random assignment to avoid bias from unmeasured confounders and instead impose assumptions that restrict the scope of the possible confounders. Specifically, DID designs assume that confounders varying across the groups are time invariant, and time-varying confounders are group invariant. Researchers refer to these twin claims as a common trend assumption. In the next two sections, we describe the DID design further and explain how the key assumptions of the design lead to a statistical modeling framework in which treatment effects are easy to estimate.
We start with the simple two-group two-period DID model and then examine a more general
design that allows for multiple groups and time periods.