Testing Causal Relationships with Interference as the Primary Motivation

Question

One of the core components of causal inference is the consistency of treatment. One element of this is the absence of interference, where the exposure in a spatial/temporal unit does not affect the outcome of a non-exposed unit. Interference can impact units spatially (treatment effects "leaking" over borders for example, in a country-level study) or temporally (anticipation effects).

However, anticipation effects are often a core point of causal inquiry. For example, I have been reading the literature on humanitarian interventions and one of the arguments levied against the use of humanitarian interventions is that the expectation of a future humanitarian intervention incentivizes dissidents to engage in asymmetric conflict with the state.

Given the implied anticipation effects, one could view this argument as a complication in testing the argument:

$Humanitarian Intervention_{it} \rightarrow Violence_{it}$

Alternatively, one could be interested in testing this anticipation effect... although I will admit that when I draw the DAG for this, it gets somewhat nonsensical because it feels like I am kind of testing the following:

$Humanitarian Intervention_{it+1} \rightarrow Violence_{it}$

So, my question is twofold: 1) with the presence of anticipation effects, is causal identification still possible, and 2) how can one test whether the anticipation effect actually exists?

Answer 1

I think the causality notation can be simplified/clarified with a toy model. Let the potential outcome equation for violence be $$V_t(H_t, V_{t-1}).$$ Violence today depends on the intervention today (negative) and violence yesterday (retaliation, so positive). Let there be four time periods: -1, 0, 1, 2. In period $t=0$, there is a policy announcement that a binary humanitarian intervention $H$ may happen in period 1 in response to $V_0$. If you like DAGs, that looks something like this:

$H$ is effective if

\begin{aligned} & V_0(H_1 = 1, V_{-1}) - V_0(H_1 = 0, V_{-1}) \\ + & V_1(H_1 = 1, V_0) - V_1(H_1 = 0, V_0) \\ + & V_2(H_1 = 1, V_1) - V_2(H_1 = 0, V_1) < 0\\ \end{aligned}

In order to simplify the notation, I've omitted the recursion. For example, the last term should really be $V_2(H_1 = 0, V_1(H_1 = 0, V_0(H_1=0, V_{-1})))$, capturing the full history.

Here the first line is the anticipation effect you are worried about. For example, the dissidents may attack more intensely now knowing that the intervention tomorrow will prevent the government from retaliating. But the government anticipates this, and jumps the gun, and so on. If the net effect is massively positive, that can reduce, offset, or worsen the positive impact of $H$ down the road. The second line is the boots-on-the-ground short-run effect (peacekeepers, diplomats, investment, aid, etc.) The third line is the long-term effect. All these quantities are well-defined, so the counterfactuals are identified.

One approach to detecting anticipation is to backdate the intervention to when individuals receive notification about their treatment assignment. If there is an effect of notification before treatment kicks in, that's evidence of anticipation.

Perhaps another way to think about this problem is that you have simultaneity between $V_0$ and $H_1$, and DAGs do not handle this naturally.