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The textbook example of a difference-in-difference regression uses time at the first difference (pre- vs. post-intervention) and geographical area as the second difference (location where intervention took place vs. location where intervention never took place).

Is there any precedence for using time for both differences and only using one location?

So, for the first difference, we would look at the months (for example) pre- and post-intervention. Then, for the second difference we would compare those to the same months of the year for the same location in a prior year.

Is this the same as DID with multiple time periods?

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In your example, you want to compare a treated entity with itself in some earlier time period. The equation is still the same and I assume you have a thorough understanding of the classical case. Here is the basic interaction model:

$$ y_{st} = α + \gamma T_{s} + \lambda P_{t} + \delta(T_{s} \times P_{t}) + \epsilon_{st}, $$

where suppose we observe some outcome $y$ in state $s$ and month $t$. Now suppose we only have one cross-sectional unit. $T_{s}$ indexes a state in a particular year. The treatment group is, for example, New York in 2020 and the control group is New York in 2019. Again, the control group is itself in the previous year. $P_{t}$ is equal to 1 during months when the intervention is in effect in both groups, 0 otherwise. Suppose the intervention goes into effect halfway through the year, so you should index all months from July onward.

In general, you should acquire a state(s) never exposed to the policy/intervention. The problem with your hypothetical approach is your post-period for a treated state is also its pre-period. I suppose a visual will help. Suppose a policy is enacted in the state of New York. You acquire monthly outcome data over several years in New York—only. Here is what I envision you wanting to do:

$$ \begin{array}{ccc} state & year & month & T & P & (T \times P) \\ \hline NY & 2019 & Jan & 0 & 0 & 0 \\ NY & 2019 & Feb & 0 & 0 & 0 \\ NY & 2019 & Mar & 0 & 0 & 0 \\ NY & 2019 & Apr & 0 & 0 & 0 \\ NY & 2019 & May & 0 & 0 & 0 \\ NY & 2019 & Jun & 0 & 0 & 0 \\ NY & 2019 & Jul & 0 & 1 & 0 \\ NY & 2019 & Aug & 0 & 1 & 0 \\ NY & 2019 & Sep & 0 & 1 & 0 \\ NY & 2019 & Oct & 0 & 1 & 0 \\ NY & 2019 & Nov & 0 & 1 & 0 \\ NY & 2019 & Dec & 0 & 1 & 0 \\ \hline NY & 2020 & Jan & 1 & 0 & 0 \\ NY & 2020 & Feb & 1 & 0 & 0 \\ NY & 2020 & Mar & 1 & 0 & 0 \\ NY & 2020 & Apr & 1 & 0 & 0 \\ NY & 2020 & May & 1 & 0 & 0 \\ NY & 2020 & Jun & 1 & 0 & 0 \\ NY & 2020 & Jul & 1 & 1 & 1 \\ NY & 2020 & Aug & 1 & 1 & 1 \\ NY & 2020 & Sep & 1 & 1 & 1 \\ NY & 2020 & Oct & 1 & 1 & 1 \\ NY & 2020 & Nov & 1 & 1 & 1 \\ NY & 2020 & Dec & 1 & 1 & 1 \\ \end{array} $$

The mechanics of difference-in-differences will give you an estimate of $\delta$, but this doesn't seem valid in my estimation. One issue is the model suffers from measurement error in $P_{t}$. Technically, your time dummy is 'turning on' in a pre-period (i.e., Jul-2019, Aug-2019, Sep-2019, etc.); it is technically New York's pre-treatment epoch. This is really a 'pre versus post' analysis in New York, with the variable $P_{t}$ indexing all months from July onward in every year.

I think your motivation for doing this is to ensure comparability across groups. What better serves as a control for New York than New York itself, right? In my opinion, I would acquire data from all other available states/counties/entities for use in such an equation. In essence, you should be comparing the pre-post differences for exposed groups with the pre-post differences for unexposed groups.

If this ad hoc technique has been used in applied work then I'm sure others will jump in and correct me. But I think applied economists would call attention to potential measurement error problems.

I hope I understood your question correctly and it helped with your intuition.

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    $\begingroup$ You are understanding my question. And I see the problem. The real reason I propose this approach is because I have no suitable control location to use. Is this specification equivalent to a pre vs. post analysis? $\endgroup$
    – Michael Webb
    Commented Nov 12, 2020 at 20:48
  • $\begingroup$ You could use this approach, but I would hesitate labeling it a difference-in-differences design. Only one group is exposed to the intervention. Right now, you're assessing the across-time change in $y$ in New York, with the across-time change in $y$ in the same jurisdiction. New York is simply an earlier version of itself, not some exclusive, unexposed unit/entity. It is, technically, a pre- versus post-treatment analysis involving one treated jurisdiction. $\endgroup$
    – Thomas Bilach
    Commented Nov 16, 2020 at 17:17

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