Five Minute Summary: Difference in Differences with a Continuous Treatment
Andrew GoodmanBacon, Pedro Sant’Anna, and I have just posted a new working paper Differences in Differences with a Continuous Treatment. In the paper, we are interested in DID strategies when a treatment variable is multivalued/continuous, and, in particular, trying to answer questions like:

What are parameters of interest in this case?

What assumptions do researchers need to make in order to identify these parameters?

Do TWFE regressions do a good job at targetting these parameters?

Can we propose alternative estimators with better properties?
This has been an exciting collaboration as I have learned a whole lot from Bacon’s work on understanding twoway fixed effects (TWFE) regressions with a staggered, binary treatment. Our (Pedro and me) earlier paper proposed some alternatives to get around the issues with TWFE regressions in this context. Our new paper has both kinds of results all in one place.
A note on terminology: what we mean by “continuous” is more like “continuous enough” that a researcher would include a single regressor in their model rather than a sequence of dummy variables. So our results are applicable to treatments such as “years of education” (which takes a fairly large number of discrete values) in addition to truly continuous treatments. Below, I’ll talk about the truly continuous case, but the paper has details about the multivalued discrete treatment case.
Baseline case with two time periods
Let’s start with the simplest case for DID where there are two time periods, no units participate in the treatment in the first time period, and some units become treated in the second time period with a continuous “dose” while other units remain untreated.
Here’s the notation I’ll use:

Two time periods: \(t\) and \(t1\)

Observed outcomes: \(Y_{it}\) and \(Y_{it1}\)

Dose (the amount of treatment that a unit experiences in the second period): \(D_i\)

In this setup, \(Y_{it} = Y_{it}(D_i)\) and \(Y_{it1} = Y_{it1}(0)\)
Parameters of Interest
With a continuous treatment, there are more potential parameters of interest than in the case with a binary treatment.
\[ATT(dd) := \E[Y_t(d)  Y_t(0)  D=d] \quad \textrm{and} \quad ACRT(dd) := \frac{\partial ATT(ld)}{\partial l}\Big_{l=d}\]\(ATT(dd)\) is the causal effect of experiencing dose \(d\) among units that actually experienced dose \(d\). \(ACRT(dd)\) is the causal response of a marginal change in the dose among units that actually experienced dose \(d\).
Parallel Trends Assumptions
With a continuous treatment, the analogue of the “standard” parallel trends assumption with a binary treatment is the following
Standard Parallel Trends: For all possible values of the dose \(d\)
\[\E[Y_t(0)  Y_{t1}(0)  D=d ] = \E[Y_t(0)  Y_{t1}(0)  D=0]\]Under this assumption (and using exactly the same arguments as in the case with a binary treatment), you can show that
\[ATT(dd) = \E[ \Delta Y_t  D=d]  \E[\Delta Y_t  D=0]\]This seems straightforward, but there is a big caveat
While \(ATT(dd)\) is the effect of dose \(d\) among units that experienced dose \(d\), most applications in economics want to answer questions like: Does more dose increase outcomes relative to less dose?
This suggests making comparisons of \(ATT(dd)\) across different values of \(d\). If we do this for two values of the dose, say, \(d_1\) and \(d_2\), notice that
\[ATT(d_2  d_2)  ATT(d_1d_1) = \underbrace{\Big( ATT(d_2d_2)  ATT(d_1d_2) \Big)}_{\textrm{causal effect}} + \underbrace{\Big( ATT(d_1d_2)  ATT(d_1d_1) \Big)}_{\textrm{selection bias}}\]In other words, comparisons of \(ATT(dd)\) across different values of the dose involve a causal effect (the average effect of moving from dose \(d_1\) to \(d_2\) among units that experienced dose \(d_2\)) but additionally involve a selection bias term (that the average effect of the treatment among units that experienced dose \(d_2\) and \(d_1\) might have been different even if they had both experienced the same dose).
The standard parallel trends assumption mentioned above is not strong enough to rule out this selection bias term because it is only an assumption about untreated potential outcomes.
This same sort of issue carries over to trying to identify causal responses: \(\frac{\partial \E[\Delta Y_t  D=d]  \E[\Delta Y_t  D=0]}{\partial d} = ACRT(dd) + \underbrace{\frac{\partial ATT(dl)}{\partial l}\Big_{l=d}}_{\textrm{selection bias}}\) In other words, derivatives of paths of outcomes also include both a causal response and a selection bias term.
This seems like a big issue.
Alternative Approaches
How can one get around the selection bias terms that show up above? We discuss a few possibilities in the paper (and it would also be interesting to think about partial identification in this context), but let’s consider here making alternative identifying assumptions. Before doing that, let’s introduce a couple of new possible parameters of interest:
\[ATE(d) := \E[Y_t(d)  Y_t(0)] \quad \textrm{and} \quad ACR := \frac{\partial ATE(d)}{\partial d}\]These are similar to \(ATT(dd)\) and \(ACRT(dd)\) above except they are causal parameters averaged across all units rather than just those that experienced dose \(d\).
Strong Parallel Trends Assumption: For all possible values of \(d\),
\[\E[Y_t(d)  Y_{t1}(0)] = \E[Y_t(d)  Y_{t1}(0)  D=d]\]Under this assumption, it is straightforward to show that
\(ATE(d) = \E[\Delta Y_t  D=d]  \E[\Delta Y_t  D=0]\) which, interestingly, is the same estimand as for \(ATT(dd)\) above, but just gets a different interpretation here because of the different assumptions. Before moving forward, it worth explicitly mentioning that strong parallel trends is likely to be a much stronger assumption than the standard parallel trends assumption. This is primarily because it involves potential outcomes for all values of \(d\) rather than just untreated potential outcomes. The payoff of making the stronger assumption is large though — it gets rid of the selection bias terms that were our main problems above. In particular, if you compare \(ATE(d)\) for different values of \(d\), you just get a causal effect term
\[\underbrace{ATE(d_2)  ATE(d_1)}_{\textrm{causal response}}\]which means that it makes sense to interpret the differences causally. Similarly, \(ACR(d)\) doesn’t include selection bias terms either under strong parallel trends.
What does all this mean? In a DID setup, if researchers want to interpret comparisons of treatment effect parameters across different values of the dose as causal effects, it likely requires substantially stronger assumptions than standard parallel trends assumptions.
TWFE Regressions
In applied work, most economists implement difference in differences with a continuous dose using a TWFE regression:
\[Y_{it} = \theta_t + \eta_i + \beta D_{it} + v_{it}\]and interpret \(\beta\) as an average causal response. In the paper, we show that
\[\beta = \int_{D_+} w_1(l) \frac{ \partial \E[\Delta Y_t  D=l]  \E[\Delta Y_tD=0]}{\partial l} \, dl + w_0 \left( \E[\Delta Y_t  D=d_L]  \E[\Delta Y_t  D=0]\right)\]where \(w_1\) and \(w_0\) are some weights that are always positive. Depending on whether one invokes standard parallel trends or strong parallel trends, the difference between expectations terms above can either be replaced by \(ACRT(ll) + \textrm{selection bias}(ll)\) or \(ACR(l)\).
That the weights are always positive is a good thing (especially compared to recent papers on binary DID with multiple periods). But the weights are still driven by the estimation method and have some strange properties. The most interesting one is that the weights are maximized at \(\E[D]\). In other words, causal responses in the middle can get substantially more weight than causal responses in the tails regardless of the distribution of the dose. This has the potential to lead to very poor estimates of the average causal response. Our suggestion is to avoid the TWFE regression here and instead just directly estimate the parameters of interest.
The Rest of the Paper…

The previous discussion seemed quite negative, but one positive thing is that, unlike the binary treatment case, you can still do DID even if there are no units that never participate in the treatment by making comparisons across units that experience different doses (since, under either parallel trends assumption, the path of untreated potential outcomes is the same and therefore cancels out when making crossdose comparisons).

Another complication is that, since the estimands under standard parallel trends and strong parallel trends are exactly the same, it is not possible to use pretests (i.e., estimate pseudo treatment effects in pretreatment periods) in order to distinguish between them.

We extend all of these results to the case with multiple time periods and variation in treatment timing.

With multiple periods and variation in treatment timing, TWFE:

Is sensitive to treatment effect dynamics (this is similar to the binary treatment case and occurs because alreadytreated units sometimes serve as controls for latetreated units in periods where the alreadytreated units treatment status does not change over time). This can lead to the negative weights issue that is prominent in the binary treatment literature.

Is additionally sensitive to heterogeneous causal responses across groups. That is, suppose that one timing group experiences more dose on average than another timing group, but that the causal response is greater for the second group than the first. This can lead to a version of the negative weighting problem. This sort of issue does not show up in the binary treatment case.

Even if you were willing to make assumptions that rule out treatment effect dynamics and heterogeneous causal responses (which would be very strong assumptions in my view), how TWFE combines different causal response parameters is still driven by the estimation method. The weights will all be positive in this case, but they could still have undesirable properties.

In my view, most importantly, besides the negative weighting issue, unless a researcher is willing to impose a substantially stronger assumption (such as a multiperiod version of the strong parallel trends assumption above), then the TWFE regression continues to contain the “selection bias” terms above.


With multiple periods and variation in treatment timing, we propose alternative estimation strategies to TWFE. These can alleviate the first three problems with TWFE mentioned above. But that, under a standard parallel trends assumption, DID includes selection bias terms is a fundamental problem in the case with a multivalued/continuous treatment. In fact, this would be
my main takeaway from our paper — in a DID setup, comparing effects across different doses likely requires substantially stronger assumptions than what most applied researchers in economics think that they are making.