Discussing Difference-in-Difference Assumption of the treatment assigment

Question

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.

Answer 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). 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.

Answer 2

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.

Answer 3

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 $Y_{it-1}$.