Difference in Difference (DD) is a commonly used empirical estimation technique in economics. Let us take a hypothetical example where a state (Wisconsin) passes a bill which makes employer-provided health insurance tax deductible. Let us also assume that in the year after the bill passed (year 2) the percentage of firms offering health insurance increased by 50% compared to the year before the bill was passed (year 1). In order to estimate the impact of the 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 50%. 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 DD regression. If there is a state (California) 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 Wisconsin and California between the two years.
We will run the regression:
Y=β_0 + β_1*T + β_2*WI + β_3*(T*WI) + e
Y is the percentage of firms offering health insurance in each state in each time period. T is a time dummy, WI is a state dummy for Wisconsin, and T*WI is the interaction of the time dummy and the Wisconsin state 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 β_0 is the baseline average, β_1 represents the time trend in the control group, β_2 represents the differences between the two states in year 1, and β_3 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.