# Heterogeneous treatment effects

## Heterogenous treatment effects

### Het DML

$$y=a(z)+d b(z)$$ Het effects is $$b(z)$$ We build groups instead of arbitrary function. So we estimate $$E[b(z)|G]$$

Use part of the data set to estimate

$$\hat y =\hat a (z)+D\hat b(z)$$

Use $$s=\hat b(z)$$ to stratify. Key point is defining subgroups algorithmically. Less opportunity for hacking

## (LATE, causal tree (from CART))

### Introduction

bart causal is different to causal tree

In stuff now two problems: + non random but constant effect + Random but heterogenous effect

causal trees can find heterogenous treatment effects

Approaches: We have treated and untreated. X and y Estimate y|x for treated, and untreated separately. Then take differnece for a given x to be the estimated treatment effect

2nd approach: have treatment as input diffence is again y|x - y|x treatment minus no treatment

3rd approach: (type of single tree) split not by predictive power, but by treatment effect differnece

4th approach: cross validation at each leaft we note the sample average treatment effect goal is to choose hyper parameters which minimise sum of diffence between these and cross valid data

Once we have the trees from the last one, calculate the effect using test data. nb: separate creating of tree to estimation of treatment effect

### Instrumental forests

Estimate LATE

like causal forest, but do IV regression on leaf.