With the groundwork laid for identifying causal effects and discovering structures from data, we now turn to quantifying these effects. This chapter specifically addresses the common challenge of estimating causal effects when faced with high-dimensional data, where the number of potential confounding variables $X$ is large. Standard statistical methods may struggle to adequately adjust for numerous covariates simultaneously.

Here, you will learn how to apply modern machine learning techniques designed to overcome these challenges and provide reliable effect estimates. We will cover:

• Double Machine Learning (DML): Implement this method to estimate Average Treatment Effects ($ATE = E[Y(1) - Y(0)]$) effectively, using machine learning to handle complex relationships between confounders, treatments, and outcomes.
• Heterogeneous Effect Estimation: Move beyond average effects to understand how treatment impacts vary across individuals or subgroups. You will learn to use Causal Forests and Meta-Learners (S-Learners, T-Learners, X-Learners) to estimate Conditional Average Treatment Effects ($CATE = E[Y(1) - Y(0) | X=x]$).
• Deep Learning Approaches: Examine how neural network architectures can be adapted for causal effect estimation tasks.
• Managing High-Dimensional Confounders: Discuss strategies for variable selection and regularization techniques tailored for causal inference in high dimensions.
• Validation of CATE Estimators: Learn methods to evaluate the performance and reliability of models that predict heterogeneous treatment effects.

This chapter provides the tools to estimate causal effects accurately in complex, real-world datasets characterized by a large number of features.