interpreting average effects from instrumental variable forest

[austindenteh]

Many thanks in advance for your time! As mentioned in #490 and discussed in this paragraph of the average treatment effect function, for an instrumental forest with binary instrument, the doubly robust average of the CLATE estimates is constructed following the paper "locally robust semiparametric estimation" by Chernozhukov et al. (2016). My questions are:

1. Is the doubly robust score implemented here the same as the score in this unnumbered equation from page 38 of the paper “Double/Debiased Machine Learning for Treatment and Structural Parameters” by Chernozhukov et al. (2018)? (see picture).

2. If the answer to (1) is "no," could you help me understand the difference between the two scores or point me an accessible reference on how this implementation follows the paper "locally robust semiparametric estimation"?
3. This paragraph of the average treatment effect function does mention an application in Section 5.2 of Athey and Wager (2021), but this assumes homogeneity such that the CATE equals the CLATE. The doubly robust score implemented here appears to use equation (44) in Athey and Wager (2021) too. Does this mean the doubly robust average for the instrumental variable forest based on the score here should only be interpreted under this homogeneity assumption?