Changing covariance logic in DRIV about econml HOT 1 CLOSED

@vsyrgkanis working through the math, it looks like all of E[(E[T|X,Z]-T)^2|X], E[(E[T|X,Z]-E[T|X])^2|X], and E[(E[T|X,Z]-E[T|X])(T-E[T|X])|X] are equal to Var(E[T|X,Z]|X) (but please correct me if I'm mistaken).

These are equivalent to training the variance model on (T_proj-T)^2, (T_proj-T_pred)^2, and (T_proj-T_pred)(T-T_pred) respectively.

It seems to me that the first of these would be noisier than the second but requires only that the projection model be well specified, while the second would be less noisy but requires both the projection and prediction models to be well-specified, and the third doesn't have any particular advantages that I can see. Do you have any thoughts on which of these approaches to use?

from econml.