Deep Learning and Kernel Methods for IV

While standard Instrumental Variable (IV) techniques offer a powerful way to address unobserved confounding, they often rely on assumptions, such as linearity, that may not hold in complex, high-dimensional machine learning settings. When the relationships between the instrument ($Z$), treatment ($T$), covariates ($X$), and outcome ($Y$) are intricate and non-linear, traditional methods like Two-Stage Least Squares (2SLS) can yield biased estimates. Deep Learning and Kernel Methods provide flexible, data-driven approaches to IV estimation, capable of capturing these complex dependencies.

Deep Instrumental Variables (Deep IV)

Deep IV adapts the core logic of 2SLS but replaces the linear models in each stage with flexible neural networks. Recall the standard IV setup where an instrument $Z$ influences treatment $T$, which in turn affects outcome $Y$. Unobserved confounders $U$ influence both $T$ and $Y$, but $Z$ is assumed to be independent of $U$ and only affect $Y$ through its effect on $T$.

The Deep IV approach, introduced by Hartford et al. (2017), consists of two main stages:

1. First Stage (Treatment Model): A neural network models the conditional distribution of the treatment $T$ given the instrument $Z$ and observed covariates $X$. Instead of just predicting the expected value $E[T | Z, X]$ like in linear 2SLS, this network often estimates the parameters of a conditional distribution $p(T | Z, X)$. For instance, if $T$ is continuous, it might estimate the mean and variance of a Gaussian mixture model. This richer representation captures the potentially complex influence of $Z$ and $X$ on $T$.

   • Let $f_{\theta_1}(Z, X)$ represent the output of this first-stage network parameterized by $\theta_1$. This output describes the conditional distribution $p(T | Z, X)$.

2. Second Stage (Outcome Model): A second neural network predicts the outcome $Y$ based on the observed covariates $X$ and the distribution of the treatment predicted by the first stage. Crucially, it does not use the observed treatment $T$ directly, as $T$ is confounded by $U$. Instead, it effectively integrates over the predicted treatment distribution:

   $$E[Y | X] \approx E_{p(T | Z, X)} [g_{\theta_2}(T, X)]$$

   where $g_{\theta_2}(T, X)$ is the second-stage neural network parameterized by $\theta_2$, representing the structural relationship between $T$, $X$, and $Y$. The parameters $\theta_1$ and $\theta_2$ are typically trained jointly by minimizing a loss function based on the final outcome prediction error, often using techniques like stochastic gradient descent.

Overview of the Deep IV framework, highlighting the two neural network stages for modeling treatment distribution and outcome prediction.

Advantages of Deep IV:

• Flexibility: Neural networks can approximate highly non-linear and complex functions, accommodating intricate relationships between $Z, T, X,$ and $Y$.
• High Dimensions: Can naturally handle high-dimensional instruments $Z$ and covariates $X$.
• Distributional Treatment: Modeling the conditional distribution $p(T | Z, X)$ is more informative than just the conditional mean, especially for complex treatment mechanisms.

Considerations for Deep IV:

• Architecture and Tuning: Performance depends significantly on neural network architecture, hyperparameter choices (learning rates, regularization), and optimization strategies. This often requires careful tuning and validation.
• Data Requirements: Deep learning models typically require substantial amounts of data for effective training.
• Interpretability: While powerful, the resulting models can be less interpretable than simpler linear IV approaches. Understanding why the model predicts a certain effect can be challenging.
• Identification Assumptions: Deep IV still relies fundamentally on the core IV assumptions: relevance ($Z$ affects $T$), independence ($Z$ is independent of $U$), and the exclusion restriction ($Z$ affects $Y$ only through $T$). The complexity of the models doesn't relax these foundational requirements.

Kernel Instrumental Variables (Kernel IV)

Kernel IV offers a non-parametric alternative using techniques from kernel methods, particularly kernel mean embeddings in Reproducing Kernel Hilbert Spaces (RKHS). Instead of explicitly modeling the first and second stages with parametric functions (like linear models or neural networks), Kernel IV focuses on satisfying moment conditions that encode the IV assumptions within an RKHS.

The core idea is to find a function $h(X)$ (representing the conditional expectation $E[Y|X, T=t]$ treated as a function of $X$ for a fixed $t$, or related quantities) that lies within an RKHS $\mathcal{H}$ and satisfies a conditional moment restriction derived from the IV assumptions. A common formulation involves solving an optimization problem akin to:

$$\min_{h \in \mathcal{H}} \sum_{i=1}^n (Y_i - h(X_i, T_i))^2 + \lambda R(h)$$

subject to constraints ensuring that the prediction errors are orthogonal to functions of the instrument $Z$ within the RKHS. This orthogonality condition is the kernel-based analogue of the requirement in linear IV that the instrument must be uncorrelated with the residual from the outcome equation.

Different Kernel IV estimators exist, often involving solving systems of equations derived from kernel matrices or employing techniques like Kernel Ridge Regression within the IV framework. For instance, Singh et al. (2019) propose methods based on kernelizing the moment conditions.

Advantages of Kernel IV:

• Non-parametric: Avoids strong assumptions about the functional forms of the treatment or outcome models.
• Theoretical Guarantees: Can offer strong theoretical convergence rates under appropriate regularity conditions.
• Handles Complexities: Effective in capturing non-linear relationships without requiring explicit specification like in Deep IV architecture design.

Considerations for Kernel IV:

• Kernel Choice: Performance depends on the choice of kernel function (e.g., Gaussian RBF, polynomial) and its parameters. This choice implicitly defines the function space (RKHS).
• Computational Cost: Kernel methods often involve operations on Gram matrices, which can scale poorly (e.g., $O(n^2)$ or $O(n^3)$) with the number of data points $n$, making them computationally intensive for very large datasets compared to stochastic gradient methods used in Deep IV.
• Interpretability: Similar to Deep IV, interpreting the resulting function $h$ within the RKHS can be challenging.

Comparison and Practical Use

• Deep IV is often favored when dealing with extremely high-dimensional inputs (images, text embeddings as instruments or covariates) where deep learning architectures excel, and when computational scalability for large datasets is a primary concern. Its performance depends heavily on architecture design and hyperparameter tuning.
• Kernel IV provides a powerful non-parametric alternative, particularly appealing when functional forms are highly uncertain and theoretical guarantees are desired. It might be more computationally feasible for moderate dataset sizes or when the dimensionality of $X$ and $Z$ is not excessive.

Both methods represent significant advancements in handling complex IV scenarios common in machine learning. They allow practitioners to move past restrictive linearity assumptions when attempting to estimate causal effects in the presence of unobserved confounders.

Implementation: Libraries like EconML provide implementations for various advanced IV estimators, including flavours of Deep IV and Kernel IV, often integrating them with other machine learning tools. Implementing these methods requires careful consideration of the underlying assumptions, model specification (network architecture or kernel choice), and validation procedures to ensure the robustness of the causal estimates.

Choosing between these methods, or standard IV, depends on the specific problem structure, data characteristics (size, dimensionality), computational resources, and the desired trade-off between flexibility, interpretability, and theoretical guarantees. Critically, the validity of the chosen instruments remains the most important factor for obtaining meaningful causal estimates, regardless of the sophistication of the estimation technique.