Techniques for High-Dimensional Confounders

Estimating causal effects using methods like Double Machine Learning (DML) or Causal Forests relies heavily on accurately modeling the nuisance functions: the propensity score $e(X) = P(T=1|X)$ and the outcome regressions $\mu(X, T) = E[Y|X, T]$. When the set of potential confounders $X = \{X_1, X_2, ..., X_p\}$ is high-dimensional (large $p$), estimating these functions using standard techniques becomes challenging due to the curse of dimensionality, potential multicollinearity, and the risk of overfitting. Simply including all $p$ variables in your models is often computationally infeasible and statistically problematic, potentially inflating the variance of your causal effect estimate. Therefore, specific strategies are required to manage high-dimensional confounders effectively within a causal inference framework.

The goal is not merely prediction accuracy for $T$ or $Y$, but identifying and adjusting for the right set of variables, those necessary to block confounding paths between treatment $T$ and outcome $Y$, without inadvertently introducing bias by controlling for colliders or mediators.

Leveraging Domain Knowledge and Causal Structure

Before resorting to purely algorithmic selection, incorporating substantive domain knowledge is invaluable. If a causal graph (even a partially specified one, based on prior research or expert opinion) is available, it can guide the initial selection of variables most likely to be confounders. Variables known a priori to be instrumental variables, mediators, or colliders under the assumed causal structure should be handled carefully, often by excluding them from the conditioning set used for adjustment.

Regularization for Nuisance Function Estimation

Regularization methods, commonly used in high-dimensional prediction tasks, can be adapted for estimating nuisance functions in causal inference. These methods introduce a penalty term to the model's loss function, encouraging simpler models and performing implicit variable selection.

Lasso (L1 Regularization)

Lasso regression adds a penalty proportional to the sum of the absolute values of the coefficients: $\lambda \sum_{j=1}^p |\beta_j|$. This encourages sparsity, meaning many coefficients are shrunk to exactly zero, effectively selecting a subset of variables.

When used for estimating $e(X)$ or $\mu(X, T)$, Lasso can help identify a relevant subset of $X$ from a high-dimensional set. In the context of DML, Lasso can be employed within the machine learning models used in the cross-fitting procedure to estimate the conditional expectations.

• Propensity Score Model: Use Lasso logistic regression (or Lasso with another appropriate link function) to model $T \sim X$.
• Outcome Model: Use Lasso regression to model $Y \sim X$ (potentially including treatment $T$ and interactions, depending on the specific DML variant or meta-learner).

Considerations:

• Lasso might arbitrarily select only one variable from a group of highly correlated confounders.
• The amount of regularization ($\lambda$) influences variable selection and coefficient bias. Standard cross-validation for tuning $\lambda$ based on predictive performance might not be optimal for causal estimation. Tuning might need to consider downstream effect estimate stability or theoretical guidance if available.

Elastic Net

Elastic Net combines L1 and L2 penalties: $\lambda_1 \sum_{j=1}^p |\beta_j| + \lambda_2 \sum_{j=1}^p \beta_j^2$. It often performs better than Lasso when covariates are highly correlated, as it tends to select groups of correlated variables together. This can be advantageous for confounder adjustment if multiple correlated variables are part of the true confounding mechanism.

Adaptive Lasso

The Adaptive Lasso applies different penalty weights to different coefficients, typically using weights derived from an initial consistent estimate (like Ridge or OLS coefficients). It possesses better theoretical properties regarding selection consistency (oracle property) under certain conditions, potentially leading to more accurate identification of the true confounders compared to standard Lasso.

Causally-Informed Feature Selection

Standard feature selection algorithms focused solely on maximizing predictive accuracy for $T$ or $Y$ can be misleading for causal inference. A variable might strongly predict the outcome but not be a confounder (e.g., a mediator), or weakly predict the outcome but be a critical confounder.

More appropriate methods aim to find a subset $W \subseteq X$ that is sufficient for deconfounding, meaning $W$ satisfies the backdoor criterion:

1. $W$ blocks all backdoor paths from $T$ to $Y$.
2. $W$ contains no descendants of $T$ (especially mediators on the causal path).

Approaches include:

• Graphical Criteria: If a reliable causal graph is estimated (see Chapter 2), algorithms can identify minimal sufficient adjustment sets directly from the graph structure.
• Algorithms Targeting Confounders: Some research focuses on algorithms specifically designed to differentiate confounders from other variable types in high dimensions, sometimes by analyzing conditional independence relationships or leveraging multiple environments/datasets.

Dimensionality Reduction: A Word of Caution

Techniques like Principal Component Analysis (PCA) or autoencoders reduce dimensionality by creating a smaller set of components or latent features that are functions of the original variables $X$. While useful for prediction, using these reduced representations directly for causal adjustment is generally problematic.

$Z = f(X)$

Adjusting for $Z$ instead of $X$ does not guarantee blocking of backdoor paths. The components in $Z$ are mixtures of the original variables, and the way they combine predictors might obscure or fail to capture the specific confounding relationships. Controlling for $Z$ can lead to biased effect estimates unless $f$ is constructed with specific knowledge of the causal structure, or under very strong assumptions. It's typically safer to perform selection or regularization on the original feature space $X$.

A simplified view where multiple observed covariates ($X_1, ..., X_p$) and an unobserved confounder ($U$) influence Treatment ($T$) and Outcome ($Y$). Selecting the correct subset of $X$ is essential for adjustment via methods like DML. $U$ represents the challenge addressed by methods in Chapter 4.

Practical Implementation and Validation

Regardless of the chosen technique (regularization, selection), implementation within frameworks like DML requires care:

1. Cross-fitting: It is essential to use cross-fitting (sample splitting) as employed in DML and Causal Forests. This prevents biases that arise when the same data is used both to select/regularize the model for nuisance components and to estimate the final causal effect. The selection or regularization process should occur within each fold of the cross-fitting procedure.
2. Sensitivity Analysis: Since the choice of method and tuning parameters (e.g., $\lambda$ in Lasso/Elastic Net) can influence the resulting effect estimate, perform sensitivity analysis. Evaluate how the estimated ATE or CATE changes under different reasonable assumptions about the confounder set or different levels of regularization. This helps assess the stability of your findings.
3. Software: Libraries like EconML provide implementations of DML and Causal Forests that readily integrate with scikit-learn models, allowing the use of LassoCV, ElasticNetCV, etc., for nuisance function estimation within the cross-fitting structure.

Ultimately, managing high-dimensional confounders requires a combination of domain knowledge, appropriate regularization or selection techniques tailored for causal estimation rather than just prediction, and careful validation through cross-fitting and sensitivity analysis. These strategies allow methods like DML and Causal Forests to yield more reliable causal effect estimates from complex, high-dimensional data.