Difference in Differences - Hypothetical Example

Hypothetical Example

Consider this example: state A passes a bill offering tax deduction to employers providing health insurance. Let us also consider that in the year after the bill passed (year 2) the percentage of firms offering health insurance increased by 30% compared to the year before the bill was passed (year 1). To estimate the impact of the bill on the percentage of firms offering health insurance, we could simply do a before and after analysis and conclude that the bill increased insurance offerings by 30%. The problem is that there could be a trend over time for more employers to offer insurance. It is impossible to identify if the tax deductibility or the time trend caused this increase in firm offering.

One way to identify the impact of the bill is to run a DID regression. If there is a state B that did not change the way it treated employer provided health insurance, we could use this as a control group to compare the changes between A and B between the two years.

We will run the regression:

where Y is the percentage of firms offering health insurance in each state in each time period. T is a time dummy, SA is a state dummy for state A, and T*SA is the interaction of the time dummy and the state A dummy.

The chart below displays the percentage of firms offering insurance in each state and time period.

The next chart explains what each coefficient in the regression represents.

We can see that is the baseline average, represents the time trend in the control group, represents the differences between the two states in year 1, and represents the difference in the changes over time. Assuming that both states have the same health insurance trends over time, we have now controlled for a possible national time trend. We can now identify what the true impact of the tax deductibility is on employers offering insurance.