This week sets the stage for the course and introduces how and why Machine Learning (ML) can be integrated into causal inference.

• Motivate the integration of statistical prediction with causal inference in response to the emergence of high-dimensional data and the need for flexible, non-linear modeling of covariates.
• Review the statistical properties of the Conditional Expectation Function (CEF) and linear regression in a low-dimensional setting.
  • The basic matrix formulation of linear regression is revisited.
• Introduce the Frisch–Waugh–Lovell (FWL) Theorem as a partialling-out technique in linear regression.
• Review asymptotic OLS inference and discuss issues with standard error estimation in high-dimensional settings.
• Summarize the concept of Neyman Orthogonality as an extension of the FWL Theorem to motivate Double Machine Learning (DML) in high-dimensional settings.

• Slides: Week 1 Introduction
• Supplements: Asymptotic OLS Inference

Optional Reading: For students who wish to explore the asymptotic properties of OLS in greater depth, see the Week 1 Supplements on asymptotic inference. Models that satisfy Neyman Orthogonality retain the classic asymptotic properties required for valid statistical inference.

Building on Week 1, where we introduced both the benefits and the challenges of high-dimensional data, this week focuses on regularization regression methods. These approaches address high dimensionality in order to improve out-of-sample prediction and strengthen statistical inference.

• Review the motivation for using high-dimensional data in analysis, and examine the limitations of ordinary linear regression in high-dimensional settings.
• Introduce regularization methods for handling high-dimensional data. We focus in particular on LASSO regression as a feature selection method under approximate sparsity, and Ridge regression for dense coefficient distributions. We also cover variants that combine LASSO and Ridge penalties.
• Introduce cross-validation and plug-in methods for fine-tuning the penalty level in regularization.
• Revisit the Frisch–Waugh–Lovell (FWL) Theorem and introduce Double LASSO for statistical inference in high-dimensional settings.
• Present other LASSO-like methods that satisfy Neyman orthogonality for valid inference.
• Demonstrate R implementations of regularization methods and Double LASSO, applying them to test the Convergence Hypothesis in Macroeconomics with high-dimensional data.

• Slides: Week 2 Machine Learning Basics
• R Code: Regularization Methods
• R Code: Double LASSO and the Convergence Hypothesis

Building on Week 2, where we introduced linear regularization methods to address high-dimensional data, this week we turn to non-linear models in Machine Learning. These approaches are designed to capture flexible and complex relationships among covariates. Our focus will be on two broad classes: Tree-based Methods and Neural Networks, along with their key variants.

• Formally introduce the concept of the bias-variance tradeoff and explain its role in tuning Machine Learning models.
• Present classic Tree-based Methods, including Regression Trees, Bagging, Random Forests, and Boosted Trees, showing how each builds on the bias-variance tradeoff.
• Introduce the foundational Neural Network framework and discuss the theoretical background of training a Neural Network model.

• Slides: Week 3 Machine Learning Advanced
• R Code: Fit ML Models

Building on the Machine Learning methods introduced in the last two weeks, this week we focus on the Double Machine Learning (DML) approach in partial linear regression, where covariates may be high-dimensional. We formally justify DML using the concept of Neyman Orthogonality, a framework that ensures consistent estimation of the treatment effect even when nuisance functions are estimated with ML. We then connect DML to the potential outcomes framework in causal inference, introducing the key assumption of conditional ignorability, which links regression-based estimation to causal interpretation.

• Formally introduce Neyman Orthogonality and explain why orthogonality is key to making ML-based nuisance estimates usable for valid inference in Double Machine Learning (DML)

• Connect DML to the partial linear regression model with high-dimensional covariates. We explain the importance of hyperparameter tuning and cross-fitting in DML and demonstrate the technique based on the high-dimensional data we used to test the Convergence Hypothesis.

• Link DML to the potential outcomes framework and conditional ignorability. We highlight how the regression-based approach ties to causal interpretation under ignorability.

• Slides: Week 4 Neyman Orthogonality and Causal Inference Basics
• R Code: Fit Double ML Models

Building on last week’s discussion of Neyman Orthogonality and the potential outcome framework, this week introduces the graphical approach to causal inference through Directed Acyclic Graphs (DAGs). We connect structural equation modeling (SEM) to modern causal reasoning by showing how DAGs formally represent causal assumptions and clarify identification strategies. This approach allows us to diagnose when and why causal effects can or cannot be identified from observational data.

We begin with a linear structural equation model to motivate the transition from equations to DAGs, demonstrating how causal paths encode relationships among treatment, covariates, and outcomes. We then introduce the backdoor criterion for identifying confounding and show its equivalence to conditional ignorability in the potential outcome framework. Next, we examine collider bias—a common pitfall that arises when conditioning on variables influenced by both treatment and outcome—and discuss the distinction between good and bad controls. Finally, we explore the front-door criterion, which enables identification even in the presence of unobserved confounders when a mediating mechanism is observed.

• Connect Structural Equation Models (SEM) to DAGs, showing how causal relationships can be represented graphically and interpreted in terms of directed paths.

• Define the backdoor criterion and explain its link to conditional ignorability as a condition for causal identification.

• Illustrate collider bias and the consequences of conditioning on colliders or post-treatment variables, using examples such as the birth-weight paradox.

• Distinguish between good controls (confounders) and bad controls (colliders, post-treatment variables, or instruments mistakenly adjusted for).

• Introduce the front-door criterion as a mechanism-based identification strategy and discuss its assumptions, advantages, and limitations.

• Slides: Week 5 Causal Inference from Directed Acyclic Graphs

This week focuses on causal inference under the Conditional Ignorability Assumption (CIA) and explores various strategies for covariate adjustment. We examine how matching, propensity score weighting, and regression adjustment provide alternative approaches for estimating treatment effects from observational data, and how these methods relate to the potential outcome framework introduced earlier.

We begin with matching and regression-based estimands, showing how both aim to eliminate selection bias by conditioning on observed covariates. We then formalize the propensity score theorem, which reduces high-dimensional covariate adjustment to a single scalar—the probability of treatment given covariates. Using this foundation, we introduce Inverse Probability Weighting (IPW) and its stabilized variant (SIPW) to construct pseudo-populations that mimic random treatment assignment.

Finally, we study Augmented Inverse Probability Weighting (AIPW), a doubly robust estimator that remains consistent if either the outcome model or the propensity model is correctly specified. We close by linking AIPW back to Neyman Orthogonality, illustrating how double robustness prepares the ground for modern Double Machine Learning approaches.

• Review the Conditional Ignorability Assumption (CIA) and discuss how matching and regression estimators address selection bias under ignorability.

• Introduce the propensity score theorem, demonstrating how causal identification can rely on a single balancing score rather than the full covariate vector.

• Derive the Inverse Probability Weighting (IPW) estimator and explore its relation to weighted regression and the Horvitz–Thompson formulation.

• Explain Stabilized IPW (SIPW) and how stabilization mitigates extreme weights and reduces estimator variance.

• Introduce G-computation as an alternative adjustment method based on outcome modeling.

• Present the Augmented Inverse Probability Weighting (AIPW) estimator, emphasizing its double robustness and connection to Neyman Orthogonality, which forms the theoretical bridge to Double Machine Learning.

• Slides: Week 6 Matching, Propensity, Weighting, and Beyond
• R Code: PSweight Demonstration

This week introduces the Instrumental Variable (IV) framework for identifying causal effects when the treatment variable is endogenous—that is, correlated with unobserved confounders. Building on the potential outcomes and DAG frameworks from earlier weeks, we explore how valid instruments can recover causal estimates by generating exogenous variation in the treatment.

We begin by revisiting the problem of omitted variable bias, showing how unobserved confounders can bias regression estimates. We then define the two key assumptions of IV identification—relevance (the instrument must shift the treatment) and exogeneity (the instrument must not directly affect the outcome). Using these conditions, we derive the Wald estimator and extend to the general Two-Stage Least Squares (2SLS) framework.

We illustrate these concepts through two canonical applications: settler mortality and institutional quality (Acemoglu, Johnson, and Robinson, 2001) and quarter of birth and education (Angrist and Krueger, 1991). We then discuss inference challenges under weak instruments, review the Anderson–Rubin (AR) test, and introduce its robust extension for use with Double Machine Learning (DML) in high-dimensional settings.

• Revisit the problem of omitted variable bias and explain why standard regression or matching approaches fail when unobserved confounding exists.

• Introduce instrumental variables and the conditions for valid identification: relevance and exogeneity.

• Derive the Wald estimator and connect it to the Two-Stage Least Squares (2SLS) procedure.

• Explore empirical applications including:

  • Colonial Origins of Comparative Development (AJR, 2001): settler mortality as an instrument for institutional quality.
  • Quarter-of-Birth and Schooling (Angrist & Krueger, 1991): compulsory schooling laws as instruments for education.

• Discuss the consequences of weak instruments and diagnose instrument strength using the first-stage F-statistic.

• Introduce the Anderson–Rubin (AR) approach to inference, which remains valid even under weak instruments.

• Extend the IV framework with Double Machine Learning for Partially Linear IV Models, combining residualization and cross-fitting for high-dimensional data.

• Slides: Week 7 Instrumental Variable Estimation
• R Code: Double Machine Learning for IV

This week introduces the Regression Discontinuity Design (RDD), a quasi-experimental method that identifies causal effects at known treatment thresholds. We begin with the sharp RDD, where treatment assignment is fully determined by a cutoff in the running variable, and extend to fuzzy RDD, where treatment probability changes discontinuously. The design’s power lies in its local randomization property near the cutoff, allowing causal inference without full ignorability.

We then cover essential estimation techniques—local linear regression, kernel weighting, and optimal bandwidth selection using the Imbens–Kalyanaraman (IK) rule—along with the interpretation of local average treatment effects (LATE). We discuss empirical examples such as Lee (2008) on electoral incumbency and Lei & Zhou (2022) on infrastructure investment thresholds.

Finally, we emphasize covariate adjustment and Machine Learning–assisted RDD. By residualizing outcomes using ML algorithms such as Random Forest, Gradient Boosting, and Lasso, we flexibly estimate nuisance functions and improve efficiency without introducing bias. This approach connects RDD to Double Machine Learning (DML) principles—using cross-fitting to ensure valid inference when high-dimensional covariates are present.

• Review sharp and fuzzy RDD frameworks and key assumptions: continuity and no manipulation.
• Derive the local average treatment effect (LATE) and implement estimation via local linear regression and kernel weighting.
• Explain the bias–variance tradeoff and optimal bandwidth selection (Imbens–Kalyanaraman rule).
• Introduce covariate control for efficiency gains, and show how nuisance functions can be estimated using Machine Learning.
• Demonstrate ML-based RDD using the Progresa/Oportunidades dataset: cross-fitted Random Forest, GBM, and Lasso residualization within bandwidth neighborhoods of the cutoff.

• Slides: Week 10 Regression Discontinuity Design
• R Code: Machine Learning for RDD

This week introduces the Difference-in-Differences (DiD) framework and its modern extensions using panel data and machine learning. We begin with the classic two-way fixed-effects (TWFE) model as the foundation for panel data analysis, showing how differencing removes time-invariant unobserved heterogeneity. We then formalize the parallel trends and no anticipation assumptions necessary for causal identification in DiD.

We illustrate how the canonical 2×2 DiD estimator arises as a special case of TWFE, and extend to staggered treatment timing, highlighting decomposition results from Goodman-Bacon (2021) and de Chaisemartin and D’Haultfoeuille (2020). We also discuss robustness issues, such as violations of parallel trends and the inclusion of lagged dependent variables to address dynamic confounding.

In the coding component, we implement Double Machine Learning (DML) for Conditional DiD, following Callaway and Sant’Anna (2021). Using county-level U.S. data on minimum wage changes (2001–2007), we estimate the average treatment effect on the treated (ATT) via cross-fitting. The DML algorithm flexibly estimates nuisance components—propensity scores and outcome regressions—using a variety of learners including Lasso, Ridge, Random Forest, and decision trees. This approach allows valid inference even with high-dimensional covariates, ensuring robustness under conditional parallel trends.

• Introduce panel data and the two-way fixed-effects (TWFE) model as a baseline for causal inference with time variation.
• Compare TWFE with the Lagged Dependent Variable (LDV) approach and discuss the Nickell bias.
• Define the Difference-in-Differences (DiD) estimator and its key assumptions (parallel trends and no anticipation).
• Connect TWFE and DiD frameworks; discuss Goodman-Bacon (2021)’s decomposition and challenges under staggered treatment timing.
• Implement Conditional DiD with Double Machine Learning (DML): cross-fitting and orthogonalization for ATT estimation using ML learners.

• Slides: Week 11 Difference-in-Differences
• R Code: DML for Conditional DiD

In these two core weeks, we move beyond estimating a single Average Treatment Effect (ATE) to uncovering Heterogeneous Treatment Effects (HTE)—how causal effects vary across individuals and subgroups. We focus on the Conditional Average Treatment Effect (CATE) framework, where the treatment effect is modeled as a function of covariates under Conditional Independence Assumption (CIA).

We begin with a review of the Interactive Regression Model (IRM) and the doubly robust formulation of the treatment effect, highlighting Neyman orthogonality and cross-fitting as key tools for valid inference when using Machine Learning for nuisance estimation. We then estimate CATEs both parametrically and nonparametrically.

In the coding component, we estimate CATEs for the effect of 401(k) eligibility on net financial assets, conditional on income. Parametric estimation uses Group Average Treatment Effects (GATEs) by income quintile and smooth polynomial approximations. Nonparametric methods employ Doubly Robust (DR) Forests and Causal Forests, which learn heterogeneity directly from data by partitioning feature space to maximize treatment effect variation. These forests produce individual-level treatment estimates with valid confidence intervals under honesty and cross-fitting.

• Introduce the Conditional Average Treatment Effect (CATE) and its identification under conditional exogeneity.
• Derive the doubly robust and orthogonal signal for estimating heterogeneous effects.
• Implement parametric estimation of CATEs (e.g., income-group GATEs and polynomial approximations).
• Introduce nonparametric estimation via Doubly Robust Forests and Causal Forests, emphasizing algorithmic honesty and cross-fitting.
• Apply these methods to estimate heterogeneous effects of 401(k) eligibility on net financial assets, showing how treatment effects increase with income.

• Slides: Week 12 and 13 Heterogeneous Treatment Effects
• R Code: DML for Conditional Average Treatment Effects