GRF Causal Forest alpha and pointJ question

[adam2392]

trafficstars

Hi,

I was using the GRF code (https://github.com/py-why/EconML/blob/8b7fe338600b7ccb6b8362f658d0ec35f5c75b7a/econml/grf/classes.py#L394-L399) and I noticed that the alpha and pointJ for CATE estimation is defined as:

• y * T
• T x T (cross-product)

I presumed pointJ was the Jacobian being estimated. This brought up some confusion for me and now I was hoping to clarify with the dev team here:

1. what is alpha in the GRF code and why is it y * T? What if T is a vector and y is just a univariate outcome? It's not documented anywhere what the behavior here is.
2. why is the point-wise Jacobian of the moment equation just the cross-product of the treatment arrays?

[vsyrgkanis]

Its documented in the parent abstract class. Every child of base grf needs to be implementing this abstract class: https://github.com/py-why/EconML/blob/8b7fe338600b7ccb6b8362f658d0ec35f5c75b7a/econml/grf/_base_grf.py#L106

[vsyrgkanis]

Note also that grf always transforms inputs to 2d matrices. Before anything. So y here is (n, 1) and T is (n, nt). This multiplication multiplies every treatment with y.

[vsyrgkanis]

The (i, j) entry of the jacobian is E[ Ti * Tj ]. So a sample of this is Ti*Tj. Each row of the pointJ is a “point” sample of the jacobian, flattened in a 1-d array. That’s why its just the cross product of the treatments

[adam2392]

Note also that grf always transforms inputs to 2d matrices. Before anything. So y here is (n, 1) and T is (n, nt). This multiplication multiplies every treatment with y.

So this transforms the weights (alpha) to be the outcome multiplied with the treatments? I can see why we might do this if T is {0, 1}, but why would you do this in general and initialize the weights as this array? Apologies if I misunderstood something.

Thanks for the quick response!

[vsyrgkanis]

These alpha is not the weights its rhe offset part of the linear moment condition that we are solving https://github.com/py-why/EconML/blob/8b7fe338600b7ccb6b8362f658d0ec35f5c75b7a/econml/grf/_base_grf.py#L45

[vsyrgkanis]

As noted rhere our GRF implementation covers only linear moment restrictions which covers almost all moments that people use frequently in practice (with the only potential exception being quantile forests)

[vsyrgkanis]

As noted rhere our GRF implementation covers only linear moment restrictions which covers almost all moments that people use frequently in practice (with the only potential exception being quantile forests)

[vsyrgkanis]

Restricting to linear moments makes computation way faster because we can simply store the local average J and the local average A (alpha) at the leaf nodes and these are sufficient statistics for all downstream calculations. We dont need to store for instance the whole “training data samples”