In Causal Inference, why is the average treatment effect usually written as $\tau = E[Y_i(1) - Y_i(0) ]$ instead of $\tau_i = E[Y_i(1) - Y_i(0) ]$?

I am wondering why in the Causal Inference literature that the Average Treatment Effect is usually written as $\tau = E[Y_i(1) - Y_i(0) ]$ instead of $\tau_i = E[Y_i(1) - Y_i(0) ]$? In other words, why is it that typically $\tau$ is NOT indexed by $i$?

Is it to be taken that $\tau = E[Y_i(1) - Y_i(0) ]$ is averaging over ALL iterations $i$?

+1 to Dimitriy's answer. To elaborate this a bit more:

The average treatment effect in the population is: $E[\tau] = E[Y^1 -Y^0] = E[Y^1]- E[Y^0]$.

We can drop the subscript $i$ not because we assume constant effect but because we assume the expected causal effect of a randomly selected individual from the population to be equal to the average causal effect across individuals in the population.

It is because $$\tau = E[\tau_i]=E[Y_i(1)-Y_i(1)],$$ that is it is the average of the individual-level treatment effects.

If the within-class distributions of the $Y_i(1)$ are identical, then

$$E[Y_i(1)] = E[Y_j(1)] \quad \forall i, j$$

• Can you elaborate a bit more? Are you saying that for what I have above, it is taking the average across all units? Like $\frac{1}{N}\sum_{i=1}^N Y_i(1) - Y_i(0)$? Feb 27 '18 at 5:04
• The $E$ stands for "expectation" - which is essentially the same as saying the mean. It's really the theoretical mean. The way I'm interpretting your question is to say that $Y_i(1)$ is observation $i$ from group 1, and $Y_i(0)$ is observation $i$ form group 0. If every observation within each group is assumed to follow the same distribution, then they have the same mean. That is, $E_{53}(1)$ should be the same as $E_{92}(1)$, and so your $\tau$ does not depend on $i$. Or if you wanted to write it that way, all $\tau_i$ would be the same. Feb 27 '18 at 17:38