# For a time-series cross-sectional (TSCS) data, how does one account for 'future' treatment history while calculating ATE/ATC/ATT?

I refer to this seminal paper (Matching Methods for Causal Inference with Time-Series Cross-Sectional Data) by Kosuke, In Song and Erik.

## Context

In the paper, the authors propose the use of a two-layer adjustment to the typical DiD estimator - (1) treatment history matching and (2) refinement methods such as PS weights / matching, Mahalanobis distance - to adjust for bias and balance covariates. The crazy thing about this proposed methodology is that one is able to have a TSCS dataset where treatment is staggered and reversed, but still be able to find the causal quantity of interest (ATT, ATC, ATE) not only contemporaneously but also for a $$T+F$$ period, where $$F$$ is the number of time periods after the baseline period $$T$$ occurs.

The authors have since also released a code implementation of this on an R package PanelMatch.

## Question

My question for those that are familiar with the paper is however this:

Say for example I am trying to understand how a sales rebate impact product sales quantity bought by a customer. In this case, $$D=1$$ when sales rebate is given for the month, and $$D=0$$ otherwise. I am interested in knowing both the contemporaneous effects and the effects in the subsequent months.

Given that treatment is allowed to be reversed, and that we allow for the calculation of causal quantity of interest for $$F$$ period (where if $$F=0$$ then contemporaneous), how do the authors account for variation in treatment statuses in the 'future' in both matched control and treated units when calculating a quantity of interest? Two examples of the variations below:

• Control unit (i.e. $$D=0$$ when $$t=T$$) switches to $$D=1$$ at $$T+f$$, where $$f? Wouldn't this affect the construction of the counterfactual 'control'?

• If treated units have different treatment status after $$t=T$$, where one treated unit may have the following sequence for $$t=[T,T+3]$$ of (1, 1, 0, 0) vs another which may have (1, 0, 0, 0)? (i.e., $$F = 3$$) wouldn't it affect the calculation of the average ATT across treated population?

The authors alluded to this under Page 10 of the paper in the following quote, but appear to chalk it up to an exception as far as I can tell:

When researchers are interested in a non-contemporaneous treatment effect (i.e. $$F>0$$), the ATT defined in Equation (8) does not specify the future treatment sequence. As a result, the matched control units may include those units who receive the treatment after time $$t$$ but before the outcome is measured at time $$t+F$$. Similarly, some treated units may return to the control conditions between time $$t$$ and time $$t+F$$.

In addition, I am also keen to understand that if there is a solution to this, whether this may have already been addressed in PanelMatch. This will be incredibly helpful to know.

Thank you all!

• Welcome. I think I need further elaboration for the first bullet. What does little “f” denote? In the second bullet, are you referring to units that “switch out” of a state of treatment? Commented Jun 27, 2023 at 5:31
• Hi Thomas, let me clarify. 1. An example to illustrate "f": If user is interested to see if there is lasting impact by the sales rebate 3 time periods after time T, then F = 3 and "f" will be [0,2]. In this case, even if a unit did not receive D at t=T (thus control), and was then matched to a treated unit, it may receive treatment at t=T+f, which I assume affects the estimation of the quantity of impact that we may be after. My question would be how did the authors actually mitigate this? Commented Jun 27, 2023 at 6:23
• 2. Yes, but more broadly, I am referring to how the authors reconcile the treatment effect across matched treatment units given that they may have very different 'future' treatment status for a given T? Commented Jun 27, 2023 at 6:27
• @ThomasBilach, I have edited the question itself to clarify the points you made - let me know if there's anything else I can help clarify, looking forward to your take on this! Commented Jun 28, 2023 at 2:23
• If I am understanding you correctly, this is probably why the authors place an emphasis on being intentional with your choice of F. As the authors state on page 7: "We note that a large value of F may make the interpretation of causal effects difficult if many units switch the treatment status during the F lead time periods." Selecting F is a bit arbitrary, but perhaps the treatment-variation plot can assist you in selecting an F so that the issues you bring up won't be problematic? As a last resort, I've emailed the authors before and they do reply to inquiries. Commented Jul 1, 2023 at 0:14

Control unit (i.e. $$D=0$$ when $$t=T$$) switches to $$D=1$$ at $$T+f$$, where $$f? Wouldn't this affect the construction of the counterfactual 'control'?

The function does not account for all possible future treatment sequences when refining the matched sets. In other words, the default behavior of PanelMatch() will allow for future policy switches. The authors already address this concern in their paper. For instance, the matched control units may include units who receive the treatment after time $$t$$ (i.e., policy switch). Similarly, some of the actual treated units may return to the control condition between time $$t$$ and time $$t + F$$. This is inevitable in settings where we want to estimate long-term causal effects.

But say you want to assess the causal effects of a rebate program where it was administered continuously over time. In that case, just be careful about your choice of $$F$$. Start by "eyeballing" the spatial-temporal distribution of the treatment variable over time; this is where the DisplayTreatment() function is your best friend. If you're interested in say the effect three months after the start of the rebate program, then most, if not all, of the treated units should remain in the treatment condition for at least 3 periods; note that as we expand the future sequence, then it's quite possible that some, or most, of the units no longer participate in the program. In essence, be careful when assessing long-term causal effects. The authors of that paper specifically caution users that a large value of $$F$$ may make the interpretation of causal effects a bit murky, especially if many units change their treatment status during the $$F$$ lead time periods.

If treated units have different treatment status after t=T, where one treated unit may have the following sequence for $$t=[T,T+3]$$ of (1, 1, 0, 0) versus another which may have (1, 0, 0, 0)? (i.e., $$F = 3$$) wouldn't it affect the calculation of the average ATT across treated population?

In short, it matters with respect to inference.

Say you specify $$F = 3$$. In the example provided, it appears that by the second lag the treatment has reversed for all units. Again, you need to be intentional with the lead = ... parameter. The values for $$F$$ should make sense given the treatment variation you're observing in the real world. Estimating short-term ATTs seems more appropriate given that treatment sequence.

I am also keen to understand that if there is a solution to this, whether this may have already been addressed in PanelMatch.

If you're concerned with the "switchers" and how they may affect inference, then proceed as recommended. Estimate the ATT of a stable policy change, where the counterfactual scenario is that a treated unit does not receive the treatment before the outcome is measured. They expound upon this alternative framework in their appendix. By employing this approach, you're saying it is not permissible for a treatment to reverse in the specified lead window. The treatment should be in place at least for $$F$$ time periods after the policy change. The specific answer you're looking for may be found in Appendix A on page 1:

We first constrain the matched set for each treated observation (i, t) such that the matched control units do not receive the treatment at least after time $$t + F$$.

But I actually prefer the explanation provided in the PanelMach documentation. When you force the treatment not to reverse in the user-defined future sequence (e.g., forbid.treatment.reversal = TRUE),

"...only matched sets for treated units where treatment is applied continuously in the lead window are included in the results."

That being said, if you know the rebate program is in place for at least a couple of months, then stick with the default settings (i.e., forbid.treatment.reversal = FALSE). It isn't fatal to observe a few treated units returning to the control condition within the specified lead window; that is a likely scenario when assessing long-term causal effects. Presently, I am working on a paper where I know the application of a policy is in effect for at least 3 months. I "know" this through domain-specific knowledge of the policy's administration and inspection of my treatment variation plot. With very few exceptions, most units return to the control condition after the third period. Given this variation, I do not impose any additional constraints on the matched sets; I'm interested in the ATT on the three periods.

On the other hand, if you're interested in the effects of "switching out" (opting out), then you may be interested in their alternative causal quantity of interest: the average treatment effect of policy reversal among the reversed (ART). Put simply, the ART is a comparison of units still in the rebate program with those that opted out (i.e., reversed). This may be worth pursuing, assuming you observe enough reversals.

In rare settings where all treated units return to the control condition at the same time, then it's difficult (if not impossible) to get matched sets for the ART. In other words, if all treated units opt out of the program by say, the fourth period, then policy reversal is deterministic and you needn't worry about it. If this is the case, then the ATT should be your preferred choice.

In sum, the best part about PanelMatch is it allows the user to define the causal quantity of interest. Since the policy variable is allowed to switch on and off multiple times over time, we should be very deliberate with our choice of $$L$$ and $$F$$. In the latter case, how far forward to go to look for policy effects is entirely up to you.

• Hi Thomas, thank you for the in-depth explanation across the scenarios laid out, this is clear to me - one should use domain knowledge to appropriately set L and F; in cases where treatment vary significantly, one should consider other metrics (e.g. ART). One follow-up question: if most units are treated for one period (with exceptions) and I am interested in the long-term (i.e. lingering) effect of the rebate, is it possible to set F to be more than 0? I assumed from my first read of the article for this to be possible, but not too if this makes sense anymore after reading your write-up. Commented Jul 3, 2023 at 3:16
• Nothing says you can't set a higher value of $F$; it will return the period-wise ATTs beyond the contemporaneous effect. In my example outlined above, I isolated a region where all units were exposed to a policy for only for a few number of periods; all units reversed the policy around the same time. I wanted to estimate some lasting effect. Well, nothing is stopping me (or you) from keeping the policy variable "turned on" for one extra period and estimating the ATT beyond the immediate adoption period. Just let your audience know you're doing this. I would give it a shot. Commented Jul 5, 2023 at 21:56
• Much appreciated Thomas, as someone who is relatively new to the topic you have made this easy to understand. Have a good day! Commented Jul 7, 2023 at 14:40
• No problem. I have enjoyed using PanelMatch. Please follow-up should you have any more questions. Commented Jul 7, 2023 at 17:21