<p>The first source you found is correct, parallel trend is not sufficient, you can find the same assumption mentioned in multiple places (e.g <a href="https://diff.healthpolicydatascience.org/" rel="nofollow noreferrer">here</a>). One of the identifying assumptions of DiD is that:</p>
<p><span class="math-container">$$Y(t)=Y_{i0}(t)=Y_{i1}(t) \quad \text{for} \quad t&lt;T_0$$</span></p>
<p>where <span class="math-container">$Y_i$</span> is the outcome of individual <span class="math-container">$i$</span>, <span class="math-container">$Y_{i0}$</span> is the potential outcome for individual <span class="math-container">$i$</span> if individual is not in treatment, and <span class="math-container">$Y_{i1}$</span> potential outcome for the same individual if the individual is in treatment arm, and <span class="math-container">$T_0$</span> treatment time.</p>
<p>If treatment is conditioned on some <span class="math-container">$Y_i(t)&gt; \hat{Y}(t)_i$</span> then it is not possible that <span class="math-container">$Y_i(t)=Y_{i0}(t)=Y_{i1}(t)$</span> in the period before treatment. Note this is not the same as requiring treatment to be randomly assigned, randomization with DiD is not required. In addition DiD allows even for strategic or confounded selection into the treatment group, but it can't be time varying.</p>

CC BY-SA 4.0
2022-12-13T07:10:51.060Z
I also thought about no anticipation, but can we really say that the treatment had a causal effect on the outcome before introduction here? The causality seems to run the other way around.
Papayapap
CC BY-SA 4.0
2022-12-13T10:28:13.103Z
@Papayapap I do believe that in this situation the potential outcomes will be different, if crime is too bad people will believe policy maker will have to do something and adjust.
1muflon1
CC BY-SA 4.0
2022-12-13T14:45:26.687Z
But it is not the policy that made crime worse. If the policy needs to be independent from the outcome, then we might as well assume exogeneity and run a normal regression Y = a + bD + e
Papayapap
CC BY-SA 4.0
2022-12-13T15:03:02.150Z
@Papayapap but that does not matter, it still violates the assumption that potential outcomes are different
1muflon1
CC BY-SA 4.0
2022-12-13T15:55:46.650Z
I don't think that's true and the direction of causality matters. Treatment does not causally affect by Y(t-1), it is affected Y(t-1). But I admit that I am not 100% sure anymore. Maybe we should open another question about it.
Papayapap
CC BY-SA 4.0
2022-12-13T16:00:02.000Z
Think about the counterfactual: if there were no policy in t, would this change Y(t-1)? Clearly not.
Papayapap
CC BY-SA 4.0
2022-12-13T16:40:41.097Z
@Papayapap if you want you can open another Q about it, now your comments make me unsure. The way how I understand it you have to ensure that future treatment status does affect your potential outcomes. The way as I see it if you for example only offer cancer drug to people who self report high levels of pain you are setting up a system in which potential outcomes pre-treatment wont be equal (I think this is good analogy for problem here). In any case, after the comments I guess impostor syndrome kicked in and I am not sure anymore. If you want you can open another Q here or on cross validated
1muflon1
CC BY-SA 4.0
2023-03-09T02:00:59.967Z
@Papayapap i asked this to Jeffrey Wooldridge. His answer is that the working assumption must be that treatment assignment does not depend on trajectory (trend) but just level differences. See the following https://twitter.com/JohnMic95688430/status/1632539546561724422?t=gf8JXyos-_oiSlJ1NuRkVw&s=19
John M. Riveros
CC BY-SA 4.0
2024-10-08T08:56:01.713Z
I am confused about your answer @1muflon1... the assumption of DiD is not that untreated outcome is the same as the treated one at every t, but that the time difference in treated outcomes is the same as the untreated one.
Matifou
CC BY-SA 4.0
2022-12-19T13:58:34.843Z
@Papayapap this is exactly the type of questions why I did the question on the first place.
John M. Riveros
CC BY-SA 4.0
2022-12-19T14:23:22.467Z
I opened another question on it but unfortunately no answer. https://economics.stackexchange.com/questions/53883/meaning-of-non-anticipation-assumption-in-panel-context 
If you have twitter, you could try asking there. The DiD crowd is quite active and will probably answer.
Papayapap
CC BY-SA 4.0
2022-12-19T15:59:31.967Z
@Papayapap I seen the question and upvoted it already for larger visibility but I think its not likely it will get answered here as there are not that many buffs when it comes to treatment evaluation here. Maybe try Cross-Validated
1muflon1
CC BY-SA 4.0
2022-12-13T00:22:24.977Z
2022-12-13T00:22:24.977Z
1muflon1
<p>You're right, we don't need random assignment but parallel trends and the two are different assumptions. Of course, if a policy were randomly assigned, parallel trends will hold. But random assignment is not a requirement for parallel trends to hold.</p>

CC BY-SA 4.0
2022-12-12T23:35:30.613Z
2022-12-12T23:35:30.613Z
Papayapap
<p>Considering that treatment assigment at time t is defined wiyj <span class="math-container">$T_{i}$</span> where it is stated this is not randomized, and the selection depends on the severity of the homicides of previous years within an informational set <span class="math-container">$\Omega_{t}=A(L)Y_{it}$</span>. Where <span class="math-container">$A(L)$</span> is an autoregressive polynomial of <span class="math-container">$Y_{it}$</span> One can state that if such information is used from the first lag only, then:</p>
<p><span class="math-container">$$ T_{it} = f(Y_{it-1})$$</span></p>
<p>Now in the generic DiD set up you have:</p>
<p><span class="math-container">$$ Y_{it} = \beta_{1} (T_{it} *t) + \mu_{i} + \lambda_{t} + u_{it} $$</span></p>
<p>Where the outcome is <span class="math-container">$Y_{it}$</span>, with a treatment dummy variable as <span class="math-container">$T_{it}$</span> and a time dummy variable as <span class="math-container">$t$</span> which if it is equal <span class="math-container">$t=1$</span> then the intervention took place and <span class="math-container">$t=0$</span> the intervention has not taken place. By merging both into a single one we have:</p>
<p><span class="math-container">$$ Y_{it} = \beta_{1} (f(Y_{it-1})*t) + \mu_{i} + \lambda_{t} + u_{it} $$</span></p>
<p>By omitting the Nickell bias raised from the autoregressive term <span class="math-container">$Y_{it}$</span>, one can obtain consistent estimates of the treatment effect in <span class="math-container">$\beta_{1}$</span> even if the treatment assignment is a function of the previous value of the outcome.</p>
<p>The only problem that threatens the causal inference is the reverse causality that may exists from <span class="math-container">$Y_{it}$</span> towards <span class="math-container">$Y_{it-1}$</span> and this is where anticipatory effects raise up relative to the intervention.</p>
<p>That is, if units are good are predicting that they're going to be part of the treatment and if is an entire decision of them and not some exogenous force, then reverse causality can create and inconsistent estimate of the treatment effect in <span class="math-container">$\beta$</span>. Hence this relies to the traditional assumptions that <span class="math-container">$T$</span> must be exogenous (in the sense that it should be given not as a decision of the units but rather an exogenous decision) and there are no anticipayory effects, because if units are able to correctly predict the future, then such function raises the reverse causality as <span class="math-container">$Y_{it}$</span> changes, will imply a change in <span class="math-container">$Y_{it-1}$</span>.</p>

CC BY-SA 4.0
2023-02-05T19:06:50.040Z
2023-02-05T19:06:50.040Z
John M. Riveros
<p>while reviewing some of the literature on DiD and Event Study designs, I come up with a general question, since DiD is by definition a quasi-experimental approach to determining causal effects when randomization of the treatment is not possible, What happens when the previous values of the outcome indirectly determine the assignment of the treatment at a certain year. The DiD stills valid?</p>
<p>Consider for example that <span class="math-container">$H_{it}$</span> is the homicide rate for some cities. Given that for some cities the homicides are so high, policy intervention is done at <span class="math-container">$t=0$</span> for the cities with the highest homicide rates, classified in the treatment dummy variable as <span class="math-container">$TD_{i}=1$</span> and <span class="math-container">$TD_{i}=0$</span> if city was not targeted by the policy.</p>
<p>Clearly this a real policy scenario where randomization is not possible, but this stills violates the DiD analysis?, I would argue it doesn't but I found some authors which state that outcome must not influenciate the treatment (<a href="https://www.publichealth.columbia.edu/research/population-health-methods/difference-difference-estimation" rel="nofollow noreferrer">https://www.publichealth.columbia.edu/research/population-health-methods/difference-difference-estimation</a>). This makes sense to me because otherwise treatment would be subject to reverse causality, but one could argue that policy intervention was done considering previous values of <span class="math-container">$H_{it}$</span>. Hence, contemporaneous outcome would not influenciate treatment assigment, and also the decision to be part of the treatment is not exactly a &quot;decision&quot; of the cities but rather based on observable past periods of the outcome, therefore, critic assumption is the parallel trends between the treated and untreated cities (also as the World Bank suggest <a href="https://dimewiki.worldbank.org/Difference-in-Differences" rel="nofollow noreferrer">https://dimewiki.worldbank.org/Difference-in-Differences</a>).</p>
<p>I would say then that DiD stills valid as long as the parallel trend assumption (and no general equilibrium effects) exists from the treatment implementation, even when the outcome somehow is related to the classification of the treatment units at least in the information of previous periods of time used by the authorities to intervene or not a city.</p>
<p>I am aware that this fits more in the Regression Discontinuity Desiggn (RDD) but in general it should also work for the DiD.</p>

<difference-in-difference><causal-inference><policy>
Discussing Difference-in-Difference Assumption of the treatment assigment
John M. Riveros
# Discussing Difference-in-Difference Assumption of the treatment assigment

(Asked by: username_0 on 2022-12-12)

while reviewing some of the literature on DiD and Event Study designs, I come up with a general question, since DiD is by definition a quasi-experimental approach to determining causal effects when randomization of the treatment is not possible, What happens when the previous values of the outcome indirectly determine the assignment of the treatment at a certain year. The DiD stills valid?

Consider for example that $H\_{it}$ is the homicide rate for some cities. Given that for some cities the homicides are so high, policy intervention is done at $t=0$ for the cities with the highest homicide rates, classified in the treatment dummy variable as $TD\_{i}=1$ and $TD\_{i}=0$ if city was not targeted by the policy.

Clearly this a real policy scenario where randomization is not possible, but this stills violates the DiD analysis?, I would argue it doesn't but I found some authors which state that outcome must not influenciate the treatment (<https://www.publichealth.columbia.edu/research/population-health-methods/difference-difference-estimation>). This makes sense to me because otherwise treatment would be subject to reverse causality, but one could argue that policy intervention was done considering previous values of $H\_{it}$. Hence, contemporaneous outcome would not influenciate treatment assigment, and also the decision to be part of the treatment is not exactly a "decision" of the cities but rather based on observable past periods of the outcome, therefore, critic assumption is the parallel trends between the treated and untreated cities (also as the World Bank suggest <https://dimewiki.worldbank.org/Difference-in-Differences>).

I would say then that DiD stills valid as long as the parallel trend assumption (and no general equilibrium effects) exists from the treatment implementation, even when the outcome somehow is related to the classification of the treatment units at least in the information of previous periods of time used by the authorities to intervene or not a city.

I am aware that this fits more in the Regression Discontinuity Desiggn (RDD) but in general it should also work for the DiD.

**Tags:** difference-in-difference, causal-inference, policy


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## 3 answers

### Answer by username_1 on 2022-12-13. Score: 1

The first source you found is correct, parallel trend is not sufficient, you can find the same assumption mentioned in multiple places (e.g [here](https://diff.healthpolicydatascience.org/)). One of the identifying assumptions of DiD is that:

$$Y(t)=Y\_{i0}(t)=Y\_{i1}(t) \quad \text{for} \quad t<T\_0$$

where $Y\_i$ is the outcome of individual $i$, $Y\_{i0}$ is the potential outcome for individual $i$ if individual is not in treatment, and $Y\_{i1}$ potential outcome for the same individual if the individual is in treatment arm, and $T\_0$ treatment time.

If treatment is conditioned on some $Y\_i(t)> \hat{Y}(t)\_i$ then it is not possible that $Y\_i(t)=Y\_{i0}(t)=Y\_{i1}(t)$ in the period before treatment. Note this is not the same as requiring treatment to be randomly assigned, randomization with DiD is not required. In addition DiD allows even for strategic or confounded selection into the treatment group, but it can't be time varying.

> **Comments:**
> **username_2**: I also thought about no anticipation, but can we really say that the treatment had a causal effect on the outcome before introduction here? The causality seems to run the other way around.  
> **username_1**: @username_2 I do believe that in this situation the potential outcomes will be different, if crime is too bad people will believe policy maker will have to do something and adjust.  
> **username_2**: But it is not the policy that made crime worse. If the policy needs to be independent from the outcome, then we might as well assume exogeneity and run a normal regression Y = a + bD + e  
> **username_1**: @username_2 but that does not matter, it still violates the assumption that potential outcomes are different  
> **username_2**: I don't think that's true and the direction of causality matters. Treatment does not causally affect by Y(t-1), it is affected Y(t-1). But I admit that I am not 100% sure anymore. Maybe we should open another question about it.  
> **username_2**: Think about the counterfactual: if there were no policy in t, would this change Y(t-1)? Clearly not.  
> **username_1**: @username_2 if you want you can open another Q about it, now your comments make me unsure. The way how I understand it you have to ensure that future treatment status does affect your potential outcomes. The way as I see it if you for example only offer cancer drug to people who self report high levels of pain you are setting up a system in which potential outcomes pre-treatment wont be equal (I think this is good analogy for problem here). In any case, after the comments I guess impostor syndrome kicked in and I am not sure anymore. If you want you can open another Q here or on cross validated  
> **username_0**: @username_2 this is exactly the type of questions why I did the question on the first place.  
> **username_2**: I opened another question on it but unfortunately no answer. https://economics.stackexchange.com/questions/53883/meaning-of-non-anticipation-assumption-in-panel-context  If you have twitter, you could try asking there. The DiD crowd is quite active and will probably answer.  
> **username_1**: @username_2 I seen the question and upvoted it already for larger visibility but I think its not likely it will get answered here as there are not that many buffs when it comes to treatment evaluation here. Maybe try Cross-Validated  
> **username_0**: @username_2 i asked this to Jeffrey Wooldridge. His answer is that the working assumption must be that treatment assignment does not depend on trajectory (trend) but just level differences. See the following https://twitter.com/JohnMic95688430/status/1632539546561724422?t=gf8JXyos-_oiSlJ1NuRkVw&s=19  
> **username_3**: I am confused about your answer @username_1... the assumption of DiD is not that untreated outcome is the same as the treated one at every t, but that the time difference in treated outcomes is the same as the untreated one.  

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### Answer by username_2 on 2022-12-12. Score: 0

You're right, we don't need random assignment but parallel trends and the two are different assumptions. Of course, if a policy were randomly assigned, parallel trends will hold. But random assignment is not a requirement for parallel trends to hold.


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### Answer by username_0 on 2023-02-05. Score: 0

Considering that treatment assigment at time t is defined wiyj $T\_{i}$ where it is stated this is not randomized, and the selection depends on the severity of the homicides of previous years within an informational set $\Omega\_{t}=A(L)Y\_{it}$. Where $A(L)$ is an autoregressive polynomial of $Y\_{it}$ One can state that if such information is used from the first lag only, then:

$$ T\_{it} = f(Y\_{it-1})$$

Now in the generic DiD set up you have:

$$ Y\_{it} = \beta\_{1} (T\_{it} \*t) + \mu\_{i} + \lambda\_{t} + u\_{it} $$

Where the outcome is $Y\_{it}$, with a treatment dummy variable as $T\_{it}$ and a time dummy variable as $t$ which if it is equal $t=1$ then the intervention took place and $t=0$ the intervention has not taken place. By merging both into a single one we have:

$$ Y\_{it} = \beta\_{1} (f(Y\_{it-1})\*t) + \mu\_{i} + \lambda\_{t} + u\_{it} $$

By omitting the Nickell bias raised from the autoregressive term $Y\_{it}$, one can obtain consistent estimates of the treatment effect in $\beta\_{1}$ even if the treatment assignment is a function of the previous value of the outcome.

The only problem that threatens the causal inference is the reverse causality that may exists from $Y\_{it}$ towards $Y\_{it-1}$ and this is where anticipatory effects raise up relative to the intervention.

That is, if units are good are predicting that they're going to be part of the treatment and if is an entire decision of them and not some exogenous force, then reverse causality can create and inconsistent estimate of the treatment effect in $\beta$. Hence this relies to the traditional assumptions that $T$ must be exogenous (in the sense that it should be given not as a decision of the units but rather an exogenous decision) and there are no anticipayory effects, because if units are able to correctly predict the future, then such function raises the reverse causality as $Y\_{it}$ changes, will imply a change in $Y\_{it-1}$.


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