Question on Inference Method & Marginal ATE vs ATE

[titubs]

@kbattocchi Hello Keith, I hope you are doing well. I wanted to understand a couple of unique properties of econml that apply to many estimators (I am looking right not at the DeepIV one https://econml.azurewebsites.net/_autosummary/econml.iv.nnet.DeepIV.html):

1. What is the difference between ATE vs ATE_INFERENCE? In the doc, ATE_INFERENCE refers to "Inference results for the quantity" but I am not sure what it means. Can you share an example?
2. What is the difference between ATE vs MARGINAL_ATE? What is a marginal effect compared to treatment effect? Thanks!

[kbattocchi]

The ate method returns just the point estimate of the ATE while the ate_inference method returns an "inference results" object which contains not only the point estimate but methods for providing confidence intervals, p-values, etc.

The ATE is the average change in the response when moving from one treatment value to another (for most of our classes these values default to 0 and 1, but DeepIV is an outlier in this regard). The marginal ATE is the derivative of this quantity at a particular treatment value, which basically says if I very slightly increase the treatment, what will be the relative increase in the response.

For estimators that compute an effect that is linear in the treatment (such as our DML estimators, among others), the ATE can be written as ϴ(T1-T0) and the marginal ATE is just the constant ϴ (regardless of the treatment value), so we provide the const_marginal_ate method which doesn't require you to specify the treatment where we take the derivative; this will also provide the exact same result as the ate method with the default treatment arguments, since ϴ(1-0) = ϴ.

[titubs]

@kbattocchi Gotcha, very helpful. Just just make sure I understood correctly for the Marginal_ATE, so basically it tells us the relative change in the outcome based on a slight change in the treatment (i.e. derivative). Do I interpret it as something like this?:

Example:

• If I change a continuous treatment by 1 unit, i.e price=$2.50 to $3.50, what is the change in outcome? Or would this refer to 1 unit as cents (i.e. from $2.50 to $2.51)?
• If I change a discrete treatment from 0 to 1 (or vice versa), what is the change in outcome?

Can you please confirm or correct me?

[kbattocchi]

Mathematically, it's the limit of ate(X,T,T+dT)/dT as dT gets arbitrarily small (this only literally makes sense for continuous treatments, but for discrete treatments you could think of it as incrementing the probability of treatment by that tiny amount).

To take your first example, the difference in outcome from moving the treatment from $2.5 to $3.5 is the ATE, specifically ATE(X, 2.5, 3.5). If we want to know the marginal ATE at $2.5, we can view this as a coarse approximation to it (we increased the price by a dollar, rather than an infinitesimal amount). We can get a closer approximation to the marginal effect if we then look at a smaller price difference, say the difference when moving from $2.5 to $2.6 (but we have to scale the result by dividing by 0.1, to make up for the difference in the range that we're looking at), or in other words, marginal_ATE(X,2.5) ~= 10*ate(X, 2.5, 2.6). Then we could get an even better approximation if we change from $2.5 to $2.51 instead of $2.5 to $2.6 (but multiply by 100 instead of 10), etc. If the effect is actually linear (which is true for many of our models), then it turns out that these will all give you exactly the same answer.