Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
57,162 characters · 10 sections · 83 citation commands
Refining the Notion of No Anticipation in Difference-in-Differences Studies
We aim to clarify a widespread ambiguity in identification arguments used in difference-in-differences (DiD) and related causal inference methods. In particular, some scholars have claimed that DiD methods require an assumption of “no anticipation” roth2023trending,callaway2021did,li2024guide,athey2022design, renson2023transporting, zeldow2021confounding,gong2021bounds,huang2023practical,baker2025,chernozhukov2024applied, TchetgenTchetgen2023,Arkhangelsky2024, Wooldridge2023. For example, Roth et al.\ roth2023trending considered this assumption to be an “often hidden assumption” which requires “that the treatment has no causal effect prior to its implementation”. Some authors have even called the no-anticipation assumption the “arrow of time” hettinger2024framework,diffindiff_resource, referring to the notion from classical physics that, macroscopically, time is perceived to flow in a one-way direction wiki2025arrow. Yet, canonical causal models are usually defined with respect to a temporal (or topological) order, where events in the future cannot affect the past. For example, causal directed acyclic graphs hernan2020causal, pearl2009causality, single world intervention graphs hernan2020causal, richardson2013swig, and conventional recursive structural equation models pearl2009causality, establish a topological order preventing future interventions to affect the past. Thus, the assumption of no anticipation appears unnecessary when one is committed to a canonical causal model. Perhaps for this reason, other works on DiD do not explicitly state the assumption abadie2005semiparametric,angrist2009mostly, Angrist1999, Caniglia2020, Meyer1995.
We claim that the confusion around anticipation effects stems from an inadequate definition of the intervention considered. This ambiguity is not only of theoretical interest: it concerns the validity and interpretation of methods that are routinely used in applied studies. We propose a solution, which relies on an expanded causal model and a new formalization of the no-anticipation assumption.
Consider first a canonical setting roth2023trending with treatment and outcome observed at two time points. Suppose that we observe independent and identically distributed draws from a near-infinite superpopulation. We consider time points $k \in \{1,2\}$ and let $A_k$ indicate whether the unit implemented the treatment ($A_k = 1$) or not ($A_k = 0$) at time $k$. At time 1, no unit implements the policy. Therefore, $A_1 = 0$ for all units. Let $Y_k$ indicate the outcome at time $k$. We are interested in studying the effect of implementing the policy at time 2, i.e., the effect of a policy implemented between the first and second measurements of the outcome. Formally, we use superscripts to denote potential outcomes hernan2020causal. Thus, $Y_2^{a_2 = a^*}$ is the potential outcome at time 2 had, possibly contrary to fact, $A_2$ been set to $a^*$. The conventional estimand of interest in DiD analyses is the average treatment effect among the treated roth2023trending:
For example, suppose that we are interested in the effect of a policy that restricts alcohol trade ($A_k$) on cigarettes sales ($Y_k$). No region implemented the policy at time 1. At time 2, the policy on alcohol restriction was implemented in some, but not all, regions. The $ATT_{A_2}$ captures the average difference in cigarette sales at time 2 among regions that actually implemented the policy, comparing outcomes under implementation versus non-implementation.
To identify the $ATT_{A_2}$, we consider standard DiD assumptions. We first state a simple positivity condition.
In our example, Assumption (ref) requires that some regions implemented the policy at time 2 and others did not. We also explicitly state a causal consistency hernan2020causal assumption.
Assumption (ref) requires that the observed cigarettes sales at time 2 for a region that implemented the policy equals the cigarettes sales had we intervened to implement the policy. Interference between regions could, for example, lead to violation of this assumption.
Finally, standard DiD involves a parallel trends condition roth2023trending.
Assumption (ref) enforces a relation between potential outcomes in those who were (factually) treated and not treated.
Under Assumptions (ref)--(ref), the $ATT_{A_2}$ is identified as
see (ref) for a standard proof, included here to emphasize that it does not rely on an additional no-anticipation assumption.
Assumptions (ref)--(ref) are sufficient for identification of the $ATT_{A_2}$. Many authors, however, explicitly state an additional assumption: the no-anticipation, or no anticipatory effects, assumption. The no-anticipation assumption is considered “important” roth2023trending, and suggested to be necessary for identification of the $ATT_{A_2}$ roth2023trending. Baker et al.\ consider the no-anticipation assumption “crucial to the validity of the DiD estimator”baker2025. To be explicit, define $Y_2^{a_1 = 0, a_2 = 0}$ as the potential outcome at time 2 had we intervened to not implement the treatment at both times, and $Y_2^{a_1 = 0, a_2 = 1}$ as the potential outcome had $A_1$ and $A_2$ been fixed to 0 and 1, respectively. Some authors gong2021bounds,huang2023practical,roth2023trending formulate the no-anticipation assumption as: \newline
Standard no-anticipation assumption. $Y_1^{a_1 = 0, a_2 = 0} = Y_1^{a_1 = 0, a_2 = 1}$ when $A_2 = 1$. \newline
Weaker versions, where the equality only holds in expectation chernozhukov2024applied, or stronger versions, where the equality holds for every unit baker2025 are also used. Some versions of the assumption are stated under interventions on $A_2$ only, i.e., corresponding to $Y_1^{a_2 = 0} = Y_1^{a_2 = 1}$TchetgenTchetgen2023,Wooldridge2023. We argue that the standard no-anticipation assumption, and its variations, are not only unnecessary when we commit to conventional causal models, but also inadequate to capture the adopters' concerns.
Taken literally, the standard no-anticipation assumption implies that we must explicitly rule out the possibility that an intervention at a later time could affect outcomes at earlier times, as if such backward causation were an actual possibility. A violation of the standard no-anticipation assumption implies that intervening at time 2 affects outcomes at time 1. This would mean, for example, that changing the natural course of history by imposing an alcohol trade restriction in year 2 can change the number of cigarettes sold in year 1.
Several authors state the standard no-anticipation assumption, or similar versions adapted to DiD with multiple time periods roth2023trending,li2024guide,callaway2021did,athey2022design,baker2025,Arkhangelsky2024, which we discuss in Section (ref). Some authors have explicitly described the no-anticipation assumption as the assumption that “post-treatment status should have no impact on pre-treatment outcomes” gong2021bounds, that “units do not respond to treatment before receiving it” ruttenauer2024twfe, or that “future treatments cannot affect past outcomes and units cannot “anticipate” receipt of treatment and, as a consequence, alter their outcomes in pre-adoption periods” li2024guide. The standard no-anticipation assumption in expectation is sometimes called “NEPT” lechner2010estimation,pronti2023water, explained in plain English as the absence of a treatment effect on pre-treatment outcomes. For example, Lechner describes this assumption as “the assumption that in the pre-treatment period the treatment had no effect on the pre-treatment population” lechner2010estimation. Since it prevents the future to affect the past, the standard no-anticipation assumption has even been denoted as a condition about the “arrow of time” hettinger2024framework,diffindiff_resource.
In conventional causal models, like Robins' Finest Fully Randomized Causally Interpreted Structured Tree Graphs (FFRCISTG) robins1986new, which covers Pearl's non-parametric structural equation model with independent errors pearl2009causality as a strict submodel, future events cannot affect the past. Richardson and Robins encode in their “Counterfactual Existence Assumption” the principle that the past cannot be affected by the future in FFRCISTG models richardson2013swig. The “Counterfactual Existence Assumption” assumption combines the assumptions known in the literature as “consistency” and “recursive substitution” richardson2013swig. Consistency alone, without invoking a recursive causal model, is not sufficient to prevent the future from affecting the past. However, consistency and the idea that the future does not affect the past are linked, as both can be derived from the definition of a set of recursive structural equations Pearl2010. Perhaps for this reason, some introductory texts argue that the no-anticipation assumption is implied by the equality $Y_1^{a_2 = 0} = Y_1$, corresponding to the consistency assumption for the outcome at time 1 callaway2024multiperoid. Similarly, other authors have defined no anticipation as the equality between the observed outcome at time 1 and the counterfactual outcome at time 1 under any treatment, i.e., $Y_1^{a_2 = 0} = Y_1^{a_2 = 1} = Y_1$, even if the first outcome measurement is instantiated before the treatment renson2023transporting,zeldow2021confounding.
On one hand the no-anticipation assumption is presented as a crucial assumption for identification, while on the other hand, it is formulated as a self-evident statement about the future not affecting the past.
There is an interesting debate about the possibility of retro-causality in quantum physics friederich2019retrocausality and some authors have engaged with the idea that interventions invoking divine acts can affect the past Leibovici2001. However, authors invoking the standard no-anticipation assumption in DiD studies explicitly require that future interventions on $A_2$ do not affect the first measurement of the outcome $Y_1$. They do not, for example, rule out that interventions on $A_2$ affect the natural value of $A_2$. This special emphasis on the first outcome measurement suggests that the standard no-anticipation assumption, along with the variations discussed in this section, is driven by a specific concern rather than by a general principle that the future cannot influence the past.
We claim that the standard definition fails to encode the conditions that the researchers intend to formalize: that is, to preclude that prior knowledge about the plan of implementing the treatment, and not the treatment itself, affects outcomes in the pretreatment period.
To fix ideas, consider the effect of Medicaid expansion on insurance coverage, which was the motivating example in Roth et al.\ roth2023trending. Let American states be the units of interest. Then, $A_2$ represents the implementation of the Medicaid expansion, while $Y_1$ and $Y_2$ represent the insurance coverage at time 1 and 2, respectively. Roth et al.\ interpreted the standard no-anticipation assumption as ensuring that “in years prior to Medicaid expansion, insurance coverage in states that expanded Medicaid was not affected by the upcoming Medicaid expansion”roth2023trending. We argue that the concern here is not that the future can affect the past, but, rather, that the decision of expanding Medicaid can affect the insurance coverage before the Medicaid expansion is actually implemented. For example, Peng found evidence consistent with the fact that insurers in Pennsylvania changed premiums “in anticipation”, that is, before the federal government approved the Medicaid expansion Peng2017. This could happen, according to Peng, because Pennsylvania presented the waiver application before the rates were due Peng2017.
Our interpretation is supported by the arguments in Roth et al.: when explaining the no-anticipation assumption they state that “units do not act on the knowledge of their future treatment date before treatment starts” roth2023trending. This statement arguably concerns the plan of implementing the policy, or the knowledge that the policy will be implemented. In contrast, the standard definition of no anticipation just affirms that a future treatment itself cannot affect the past roth2023trending.
To formally represent the plain language notion of no anticipation, we introduce a variable $P$, representing the decision (or plan) of implementing the policy. To simplify the presentation, let $P = 1$ if there is a plan to implement the policy at time 2, and $P = 0$ otherwise. In the Medicaid example, a researcher could conceptualize $P$ as indicating whether the state submitted a waiver application to expand Medicaid. We assume that $P$ deterministically causes $A_2$:
In words, when a decision about the policy is made, the decision is always executed. In the Medicaid example, if we interpret $P$ as the waiver application, Assumption (ref) states that Medicaid expansion occurs if and only if the state submits a waiver application: if the application is submitted, the expansion is always approved; if it is not submitted, the expansion does not occur. Assumption (ref) is reasonable when $P$ represents the decision of a parliament or legislative body in the region, and the decision is to implement a policy in the next years or starting at a specific date in the future.
In certain settings, the random variable $P$ can temporally precede the first measurement of the outcome. In the Medicaid DiD study, for example, this could happen if $Y_1$ represents a measurement of insurance coverage for a period of time following the waiver application ($P$). We will focus on settings where it is possible to conceive the existence of a deterministic decision variable $P$, and in which $Y_1$ temporally follows $P$. Yet there exist settings where $P$ is implausible or redundant. For example, when the treatment of interest is a viral outbreak park2022, it may be hard to imagine a deterministic variable causing the treatment; or when there is no temporal lag between the treatment decision and treatment implementation, the distinction between $P$ and $A_2$ is unnecessarily convoluted. However, we will argue that $P$ is particularly relevant in settings where authors are concerned about anticipation, for example in DiD studies evaluating the effects of new policies, like Medicaid expansion, that are typically announced prior to their implementation.
Let $U$ represent possibly unmeasured common causes of $P,\, Y_1$, and $Y_2$. We can encode the causal structure, including $P$, in a causal directed acyclic graph representing a FFRCISTG model robins1986new,richardson2013swig (Figure (ref)a). The expanded causal model makes explicit that we can conceive of two possible distinct interventions: interventions on $P$, the decision to implement the policy; or on $A_2$, the policy implementation itself. We represented the intervention on the variable $A_2$ in Figure (ref)b, on the variable $P$ in Figure (ref)c, and on both $A_2$ and $P$ in Figure (ref)d, using single world intervention graphs richardson2013swig.
Next, we argue that the mismatch between the standard no-anticipation assumption formulation and its intended meaning is due to a failure to distinguish between interventions on $P$ and interventions on $A_2$.
Researchers who raise concerns about anticipation effects are typically concerned that the plan to implement a policy influences outcomes even before the policy is enacted. Now consider an alternative definition of no anticipation that explicitly incorporates the variable $P$, representing the decision.
Here, $Y_k^{p=1}$ is the potential outcome at time $k$ had we intervened to set $P=1$. Assumption (ref) states that there is no average effect of the decision $P$ on the first outcome measurement $Y_1$, for units that decided to implement the policy ($P = 1$). In the example on Medicaid, Assumption (ref) requires that the plan to implement the Medicaid expansion ($P$) does not affect the insurance coverage in the years before the expansion is actually implemented ($Y_1$), for states that planned the expansion. Therefore, for example, the waiver application cannot change the behavior on the insurance market for states that applied for the federal approval.
The standard no-anticipation assumption rules out that the future treatment itself affects past outcomes, while our formulation rules out that the decisions to implement the policies affect the outcomes. Differently from the standard no-anticipation assumption, Assumption (ref) is non-trivial in a FFRCISTG model. Behavioral changes before the policy implementation can be responses to policy decisions, even before the policy is implemented. Because many policy decisions are public and known in advance, units may adjust their behavior based on them.
Relatedly, Abbring and van den Berg Abbring2003 write that Robins's causal model relies on a consistency assumption ensuring that future treatments cannot affect the past Abbring2003. However, Abbring and van den Berg Abbring2003 add that this assumption, standard in biostatistics, is not always suitable for social sciences, as “in social sciences, individuals may act on knowledge of the moment of realization of a future treatment” Abbring2003. They therefore propose a no-anticipation assumption Abbring2003 on the hazard scale, conceptually similar to the standard no-anticipation assumption. The concern, stated in plain English, is that units act on existing knowledge; however, the formal assumption, stated in mathematical terms, encodes the impossibility of future treatment to affect the past.
Similarly, when discussing the example of a DiD study to assess the effect of Medicaid expansion on mortality, Baker et al.\ are concerned about policy announcements affecting pre-implementation outcomes baker2025. Their no-anticipation assumption instead just rules out an effect of the Medicaid expansion on the past. Similarly, Lechner writes that the NEPT assumption “would be violated if individuals decided not to search for a job because they know (or plausibly anticipate in a way not captured by X) that they will participate in an attractive training programme” lechner2010estimation. Again, the concern, stated in plain English, is that units act on existing knowledge; however, the formal assumption, stated in mathematical terms, encodes the impossibility of future treatment to affect the past.
These considerations and examples suggest that $A_2$ and $P$ are conflated: when discussing anticipation, authors use the symbol for treatment implementation, our $A_2$, but tell stories about implementation plans, $P$. While the precise reasons for the authors’ concern about $P$ influencing pre-implementation outcomes are not explicitly stated, we believe there are at least two possible motivations. First, they are interested in the $ATT_{A_2}$, but implicitly interpret the parallel trends assumption to be defined with respect to the intervention setting $P=0$. Second, they might be interested in the effect of an intervention on $P$, and not $A_2$. We elaborate on these two reasons in Sections (ref) and (ref), respectively.
Assumption (ref) reflects the classical parallel trends assumption roth2023trending. Consider now a parallel trends assumption with respect to $P$, rather than $A_2$:
Assumption (ref) states that the trend would have been parallel in units with observed $P = 1$ and $P = 0$, had we forced a decision of not implementing the policy. Under the determinism in Assumption (ref), Assumption (ref) means that the trends would have been parallel if the treatment group never decided to implement the policy. For example, according to Assumption (ref) states that did not apply for the Medicaid waiver and those that did should have followed the same trend in insurance coverage, had the latter not applied.
Consider also the following consistency assumption, concerning interventions on $P$.
Next, suppose that the decision $P$ only affects $Y_2$ through the mediator $A_2$. To formalize this exclusion restriction, we consider a joint intervention on the policy decision and the policy implementation, and use $Y^{p=p^{*},\, a_2=a^*}_2$ to indicate the outcome at time 2 when we set $P=p^*$ and $A^{p=p^*}_2=a^*$. This joint intervention is represented in the single world intervention graph in Figure (ref)d and the exclusion restriction is formalized as follows:
Assumption (ref) states that the decision to implement a Medicaid expansion does not affect insurance coverage at time 2, other than through the Medicaid expansion itself.
See (ref) for a proof. Conceptually, suppose that the effect of $P$ on $Y_2$ is entirely mediated by $A_2$, and $P$ and $A_2$ are deterministically related. Then, under the parallel trends with respect to $P$, the classic DiD functional in Equation (ref) is equal to the $ATT_{A_2}$ minus a bias term $\psi$. This bias term $\psi$ is exactly the anticipation effect, that is, the effect of $P$ on $Y_1$ among units that planned the treatment. Therefore, when there is no anticipation (Assumption (ref) holds), $\psi=0$ and the classic DiD functional identifies the $ATT_{A_2}$.
The concern that units adjust their behavior in response to the policy plan is reasonable if researchers believe in Assumption (ref). Proposition (ref) says that, under the stated assumptions, the classic DiD functional captures two different effects: the effect of interest ($ATT_{A_2}$), and the anticipation effect ($\psi$). This result aligns with informal statements from the literature, for example, Roth et al.\ state that if anticipation is present “changes in the outcome for the treated group between period 1 and 2 could reflect not just the causal effect in period t = 2 but also the anticipatory effect in period t = 1” roth2023trending.This quote is compatible with the idea that researchers, perhaps implicitly, interpret the parallel trends assumption under intervention $P = 0$ (Assumption (ref)), rather than the more commonly invoked parallel trends assumption under $A_2 = 0$ (Assumption (ref)).
In the next section, we discuss another possible reason for being concerned about anticipation.
The $ATT_{A_2}$ represents the effect of intervening on $A_2$, among units who naturally would be treated (observed to have $A_2 = 1$). This effect can be different from an effect of the policy decision, i.e., intervening to change the value of $P$. The single world intervention graphs in Figure (ref)b and (ref)c represent the two hypothetical interventions.
To formalize the intervention on the decision, consider the following estimand:
The $ATT_P$ represents the average treatment effect among the treated, where the treatment is now the variable $P$ and not, as in Section (ref), $A_2$. In the Medicaid example, the $ATT_P$ would correspond to the effect of deciding to expand Medicaid among states that in fact decided to expand. To further highlight the conceptual difference between $ATT_P$ and $ATT_{A_2}$, consider the following didactical example.
Some researchers seem to be interested in the $ATT_P$, rather than the $ATT_{A_2}$, when they believe anticipation is present. For example, Baker et al.\ noted that the announcement of Medicaid expansion could affect mortality before the policy was actually rolled out baker2025. They claimed that in this case the “treatment” starts at the time of the announcement, not at the time of implementation baker2025. Thus, their “treatment” arguably refers to the decision of expanding Medicaid, not the expansion itself, which occurred later. Similarly, others have proposed to redefine treatment as the announcement of a decision, in order to “immunize” the DiD analyses against anticipation effects deChaisemartin2022. Defining treatment at the time a policy is announced is a common approach to circumvent issues related to anticipation Cerulli2022. This ensures that outcomes occurring before the announcement cannot be influenced by the policy decision. This implies two things. First, it suggests that the authors do not genuinely believe that future events can affect the past; that is, they implicitly take for granted that announcements do not affect the past. Second, it reveals that the authors regard the effect of the policy decision itself, as opposed to its implementation, as the causal effect of interest.
In the setting with two time points presented in Section (ref), when the no-anticipation Assumption (ref) holds and $P$ is the treatment of interest, $ATT_P$ is identified by the classical DiD functional:
See (ref) for details about the proof.
Informally, classical DiD methods target the $ATT_P$ if there is no-anticipation, $A_2$ is deterministically equal to $P$, and parallel trends with respect to an intervention fixing $P=0$ holds. Under the assumptions invoked in Proposition (ref), the $ATT_{A_2}$ and the $ATT_{P}$ can still differ. However, when the determinism in Assumption (ref) and the full-mediation condition in Assumption (ref) hold, the two estimands coincide.
As discussed in Section (ref), the interventions we consider on $P$ and $A_2$ are related to the interventions defining separable effects stensrud2022conditional, stensrud2021generalized, martinussen2023estimation, richardson2010alternative, richardson2020interventionist, wen2024causal,park2024proximal. However, our identification arguments differ from those previously invoked in separable effect settings. This is because there is unmeasured confounding between $P$ and $(Y_1, Y_2)$, violating the classical separable effects conditions richardson2020interventionist,stensrud2021generalized, like the dismissible component conditions. Also unlike the settings in Wen et al.\ wen2024causal and Park et al.\ park2024proximal, we do not have a perfect frontdoor variable or proxy variables. Instead, we rely on the parallel trends assumptions for identification. However, similar to the separable effect settings, we consider certain no-direct-effect assumptions, or exclusion restrictions, that allow us to equate different types of causal effects.
In the next section, we extend the results presented in Proposition (ref) to the multiple times case.
A popular extension of the DiD methodology allows for staggered policy adoption and multiple outcome measurements roth2023trending,callaway2021did. In this setting, a “limited treatment anticipation” assumption callaway2021did or a no-anticipation assumption roth2023trending are often invoked. Like the conventional assumptions for two time periods, these assumptions involve statements about the future not affecting the past.
We will specifically consider the DiD setting from Callaway and Sant'Anna callaway2021did, with measurements of treatments and outcomes over time. To make the exposition as simple as possible, we give identification arguments without conditioning on baseline covariates. However, the results can be readily adapted to the settings where DiD assumptions are conditional on baseline covariates.
As in the canonical setting, we observe units that are independent and identically distributed draws from a near-infinite superpopulation. We consider time points $k \in \{1, \ldots, \tau\}$ and indicate with $A_k$ whether the unit implemented the treatment ($A_k = 1$) or not ($A_k = 0$) at time $k$. Further, we indicate with $\bar{A}_k$ the vector of all treatment measurements from time 1 to time $k$. The outcome at time $k$ is denoted by $Y_k$. We also define the potential outcome $Y_k^{\bar{a}_\tau}$, corresponding to the value of the outcome at time $k$ if the treatment was fixed to be $\bar{A}_\tau = \bar{a}_\tau$. We use $\bar{x}_d$ with $x \in \{0,1\}$ to indicate a vector of length $d$ with elements all equal to $x$. For example, we use $Y_k^{\bar{a}_\tau = \bar{0}_\tau}$ to indicate the outcome at time $k$ had the policy never been implemented until the end of the study.
A common estimand of interest in DiD analyses with multiple time points is the “group-time average treatment effect” callaway2021did:
This effect contrasts the expected outcomes at time $k$ among units that start treatment at time $g$ between two strategies: treat from time $g$ onwards, and never treat.
To identify this estimand, Callaway and Sant’Anna invoked a “limited treatment anticipation assumption”, imposing that future interventions do not affect measurements before a certain point in time in the past callaway2021did: \newline
Standard limited treatment anticipation assumption. There is a $\delta \geq 0$, such that \[ \mathbb{E}(Y_k^{\bar{a}_\tau = \{ \bar{0}_{g-1}, \bar{1}_{\tau - g + 1} \}} \mid \bar{A}_\tau = \{ \bar{0}_{g-1}, \bar{1}_{\tau - g + 1} \}) = \mathbb{E}(Y_k^{\bar{a}_\tau = \bar{0}_\tau} \mid \bar{A}_\tau = \{ \bar{0}_{g-1}, \bar{1}_{\tau - g + 1} \}) \] for all $g, k$ such that $k < g - \delta$.\newline
For example, if $\delta = 1$ this assumption allows that an intervention at time $g$ affects the outcome at time $g-1$. In other words, events in the future can affect events in the (recent) past.
While we ground our presentation on the important work of Callaway and Sant’Anna callaway2021did, other definitions of no anticipation also exist for the multiple time periods case:\newline
Standard staggered no-anticipation assumption. $Y_k^{\bar{a}_\tau = \{ \bar{0}_{g-1}, \bar{1}_{\tau - g + 1} \}} = Y_k^{\bar{a}_\tau = \bar{0}_\tau}$ when $k < g$.\newline
The Standard staggered no-anticipation assumption is often used in the literature roth2023trending,li2024guide,athey2022design,baker2025. The assumption states that, before time $g$, a unit that is forced to start treatment at time $g$ has the same outcome that it would have had if it was forced to never start treatment; stating that interventions in the future cannot modify outcomes in the past. A stronger version equating potential outcomes at time $k$ for all possible interventions that do not differ until time $k$ also exists Arkhangelsky2024.
As discussed in the simple setting with two time points, we believe that these counterfactual statements do not match the substantial interpretation attributed to them. To make this explicit, we consider an expanded causal model and a different estimand that represents the effect of the decision of implementing the policy, rather than the effect of implementing the policy. As in the previous sections, we will rely on a FFRCISTG causal model robins1986new, richardson2013swig.
Consider a binary variable $P_k$ indicating whether the policy makers have decided, at time $k$ or before, to implement ($P_k = 1$) or not ($P_k = 0$) the policy. Once the decision is first made, this variable will always take the value 1.
At the beginning of the study, all units are untreated and no units planned to implement the policy before time 1. However, when the decision of implementing the policy is made, the policy enters into force after $s \geq 1$ time points and remains in place until $\tau+s$.
Assumption (ref) only enforces a determinism between decision and lagged implementation in the data. Since we are interested in the effect of interventions on $P$, this assumption will suffice for our identification argument.
We indicate with $\bar{P}_k$ the history of all measurements of this variable from time 1 to time $k$. We will assume that there is a positive probability of first making the decision to implement the policy up to time $\tau - s$. For simplicity, we will assume that if, by time $\tau - s$, a unit decided to not implement the policy, it will persist in the decision to not implement the policy until the end of the study. This restriction is motivated by the fact that in some settings the decision variable is not directly observed, and it is not possible to use $\bar{A}_\tau$ to infer the value of $P_k$ for units with $P_{\tau-s} = 0$ and times $\tau - s < k \leq \tau$.
Similarly to the approach described in Section (ref), we conceptualize an intervention on the variables $P_k$ and use the potential outcome notation $Y_k^{\bar{p}_\tau}$ to indicate interventions on the vector of values of $P_k$ until time $\tau$. This motivates a new estimand for $g \leq \tau - s$ and $k \leq \tau$:
The $ATT_{\bar{P}}(g,k)$ represents the effect, at time $k$, of first deciding at time $g$ to implement the policy versus deciding during the study period to not implement the policy, among units that first decided to implement the policy at time $g$. This estimand is conceptually similar to the “group-time average treatment effect” proposed by Callaway and Sant’Anna callaway2021did,callaway2024multiperoid reported in Definition (ref), except that we are now considering a different intervention, which is the intervention on the decision rather than on the policy implementation.
We will also invoke a consistency assumption, ensuring correspondence between potential outcomes and observed outcomes.
Furthermore, we will invoke the following no-anticipation assumption.
Assumption (ref) requires that, among units deciding at time $g$ to implement the policy, there is no difference in the expected outcomes between the treatment strategy “decide at time g to first implement the policy” and “decide during the study to not implement the policy”, as long as $k<g+s$. Informally, this means that the decision to implement the policy at time $g$ does not affect the outcomes until the implementation actually takes place ($k=g+s$). A trivial implication of Assumption (ref) is that for $g \leq \tau - s$, and $k \leq \tau$ such that $k<g+s$, $ATT_{\bar{P}}(g,k)=0$.
We will finally invoke a parallel trends assumption between units that decided at time $g$ to implement the policy and units that decided to not implement the policy until later.
Here, we consider the parallel trends to hold with all units who decided not to implement the policy from time $k$ to the end of the study. By narrowing the range for $j$, it is possible to reduce Assumption (ref) to either of the parallel trends assumptions in Callaway and Sant’Anna callaway2021did using the decision as the treatment of interest. Under these assumptions, the $ATT_{\bar{P}}(g,k)$ is identified by a functional similar to the one proposed by Callaway and Sant'Anna callaway2021did.
See (ref) for the proof. This result is a direct generalization of the scenario discussed in Section (ref), where $\tau=2$ and $s=1$. In this setting, the $ATT_{\bar{P}}(1,2)$ corresponds to the functional in Proposition (ref).
We assumed that $s \geq 1$. However, if $s = 0$, the decision and the implementation occur at the same time. The functionals in Proposition (ref) when $s = 0$ become identical to the functionals from Callaway and Sant’Anna; the $ATT_{\text{unc}}^{\text{ny}}(g,k;\delta=0)$ and the $ATT_{\text{unc}}^{\text{nev}}(g,k;\delta=0)$ callaway2021did,callaway2024multiperoid. Callaway and Sant’Anna also allow $\delta > 0$ to indicate a degree of “anticipation”, see the Standard limited treatment anticipation assumption callaway2021did. In our setup this would correspond to allowing Assumption (ref) to hold only for $k$ such that $g \leq k < g + s - \delta$. As the authors point out, however, this then requires the parallel trends assumption to be true also for time points before $g + s$ (i.e., for $k$ such that $k \geq g + s - \delta$) callaway2021did.
We have described an ambiguity in the identification argument used in many DiD studies. This ambiguity concerns the assumption of no anticipation, and specifically the mismatch between the mathematical formulation of the assumption and the plain language description of its meaning. The problem stems from an unclear definition of the intervention being considered: whether it concerns the implementation of the policy or the earlier decision to implement it.
To address this, we suggest explicitly including the decision variable in the causal model. This variable can then be used to make explicit the estimand of interest, and clarify the assumptions needed for identification.
Finally, while we have focused on difference-in-differences methodology, the ambiguity of the no-anticipation assumption is not unique to this design. For example, the standard staggered no-anticipation assumption has been invoked in the context of synthetic controls BenMichael2021. No anticipation is considered a crucial assumption in methodologies leveraging time variation Abadie2021.
Marco Piccininni and Mats J. Stensrud were supported by the Swiss National Science Foundation (project funding, grant number: 207436).