In an attempt to evaluate a potential U.S. National Sales tax rate, it was suggested on this forum before that I consider a difference-in-differences on state sales tax rates before and after they changed. How would I go about doing this? Would I do a difference-in-differences on each of the states sales taxes, and then aggregate the data? If so, how do I aggregate the data?
The main value in actually setting it up as a distinct diff in diff vs just using a dummy variable for the policy implementation is that you can do a visual check of the trajectories through time of the treated group (or treated object) and the untreated up until the date of the treatment. If the two have parallel trajectories before the treatment, then they represent good diff-in-diff comparisons. Researchers who work with diff in diff will always be impressed with good parallel trajectories. It means the two (or the two groups) are moving together and the time-dependent shocks are affecting the groups similarly (no group-dependent, time-dependent shocks). Then you have a vertical line on the graph showing the time of the treatment and the curves will diverge (ideally in a step fashion or delayed-step fashion, unless the treatment effect is increasing with time).
However, any diff in diff is technically just a regression with a dummy variable for the treatment (and dependent variable that is in a time-step where it is in the treated state has a 1 for that dummy variable on the RHS), especially if you want to get p-values.
In your case you have multiple changes at varying times and of varying magnitudes, so I wouldn’t think of it, nor present it, as a diif-in-diff (except maybe as below). In addition to multiple states and times for the changes, you don’t have an “on/off” type of treatment so you don’t have a dummy variable. If you want to motivate that you have good comparisons and no endogeneity issue, there is a way to use any states that had no changes during the study period and break the other states into groups to show trajectories. (I’ll mention that below). Or even create “synthetic comparisons” if you want to get real serious and read about those. A synthetic comparison state is created using weighted sums of the other states, although that is better suited for assessing the impact on a state. A famous paper to first do this used a weighted sum of comparison states to create a “synthetic California”, if you want to google that, and showed it had parallel and even out-of-sample parallel behavior compared to California but both still before the change. I dont remember the name you can probably find it.
The state gdp can be the LHS, and on the right you have deviation from starting sales tax rate (from starting value for that state) as a continuous variable. Also include lagged deviation (past increases in sales tax, change from one and two years prior to the state-year being considered. The other benefit is that you can include any observables in the regression.
So y_t = gdp for year t = a + beta*(increase in tax rate from year-zero value) + beta2*(increase of year t-1 from year-zero value) + beta3*(increase of year t-2 from year-zero value) + other stuff.
If you you are at a pretty high-level or even snobby university or institution that turns their nose up at “regression runners” claiming you can’t control for unobservables and don’t know if you have endogeneity issues, you can break the study period into pieces. Eg: one group had changes in years 7-10 but not 0-7 or 11-15, and one group had no increases years 0-15 (sort that way and ignore any who dont fit either group). Plot the gdp mean of each group from years 0-7 to show good comparisons (parallel trajectories) and emphasize this shows we dont have any time-dependent, group-dependent shocks that will throw off the estimates. Then add the rest of the years with years 7-10 kindve either dotted-lines for the two curves, or a greyed out band, as this is transition time. The change (hopefully a step or a lagged step) between the groups shows visually the effect of sales tax. You still do the regressions and just use the coefficients from that; all this other is just to show the coefficients are measuring what you say they are. I’ll answer any questions if Im around this might be confusing.
A much simpler version and way less accurate version would use states that had no sales tax, if enough exist, but you still have the problem of different years of implementation. If you had three such states you could do three totally separate diff-in-diff tests. Let me know if you have such an unlikely situation.