Lecture 08

• Part 1: Theoretical Framework of Causal Inference
• Part 2: Classic Causal Inference Models
• Part 3: Post-selection Inference
• Part 4: Generalization and Methodology
• Part 5: High Dimension and Endogeneity
• Part 6: Going Further
• Part 7: Inference on Heterogeneous Effects
• Part 8: Optimal Policy Learning
• Part 9: Literatures

• Importance: Integrating machine learning with causal inference to enable credible estimation of treatment effects in high‑dimensional, nonlinear economic and financial data — improving bias reduction, uncovering heterogeneous effects, and supporting robust policy and decision analysis.

• Focus:

  • Potential outcomes framework and identification conditions (SUTVA, ignorability, overlap).
  • Classical causal methods: regression adjustment, propensity‑score matching, IV/2SLS, DiD, RDD.
  • ML‑augmented causal tools: Lasso (variable selection), post‑selection inference, double‑selection; Double/Debiased Machine Learning (cross‑fitting, orthogonal scores); Causal Trees and Causal Forests for heterogeneous treatment effects; TMLE for targeted estimation.
  • Practical concerns: model selection, sparsity, diagnostic checks (balance, instrument validity, parallel trends), and sensitivity analyses.

• Let:

  • : potential outcome without treatment.
  • : potential outcome with treatment.

• Observed outcome:

• Individual Treatment Effect:
  • unobservable, thus cannot be estimated
• Average Treatment Effect (ATE):

• Average Treatment Effect on Treated (ATT):

• RCTs: Random assignment to treatment/control groups.
• Control group serves as a baseline for measuring treatment effects.
• Aims to facilitate comparison with counterfactual scenarios (outcomes without treatment).

• Reduces selection bias, ensuring comparability between groups.
• Helps in establishing credible causal relationships by controlling external factors.

The assumption expresses the independence of treatment with respect to potential outcomes, which is credible if and only if treatment is randomly assigned, without reference to the individual's potential outcome.

By comparing the average outcomes between the treated group and the control group, we have:

where denotes the (average) treatment effect.

Using the central limit theorem we obtain that the estimator is asymptotically normal:

where

and we can derive confidence intervals on the ATE.

Assumptions:

1. SUTVA: The treatment effect on one individual does not affect others.
2. Random Assignment: Treatment assignment is independent of potential outcomes.

Estimation Approach:

• Linear Model:

• OLS can estimate average treatment effects under standard assumptions.

Key Assumption

• Conditional Independence:

Propensity Score

• Defines the likelihood of treatment given observable characteristics

• The propensity score allows us to reduce the dimension of the problem, since it satisfies:

• conditional on the propensity score , treatment assignment is independent of the potential outcomes.

• to estimate the score p in a first step using non-parametric regression
  • when is discrete and takes values , a natural estimator of p is when
  • Then, we use:

  • Define the oracle estimator which is obtained assuming the propensity score is known:

  • the asymptotic properties of proceeds by decomposing

For each :

where for , and thus

One way to obtain a consistent estimator of the ATE with this characterization would be to use a consistent non-parametric estimator of and then average over the observations:

The augmented inverse propensity score (AIPW) estimator, defined by Robins et al. (1994) and Hahn (1998), is designed to correct the bias in due to the estimation of . It is given by

This AIPW estimator has two important properties:

• It achieves the semiparametric efficiency bound (Robins et al., 1994; Hahn, 1998), with

  • where for .

• It is doubly robust, meaning that it is consistent either if the estimators for are consistent, or if is consistent.

• We are in a context where but not .

• Strategy: identify instrumental variables , where is larger than the number of endogenous variables (here: is scalar).

• Instrument definition:

  1. Correlated with the endogenous variable .
  2. Uncorrelated with the residuals .

• Define as the best linear prediction of using :

• Denote .

• Formal Assumptions

  • Assumption (Rank condition). is non-singular.
  • Assumption (Exogeneity).

• Identification Result

  • The true parameters take the form:

• Relevance (Lower-Level Conditions)

  • Lower-level conditions for the relevance of for :
    • is non-singular.
    • .

• The estimator obtained from the empirical counterpart of the identification formula is the two-stage least squares estimator.

• 2SLS procedure:

  1. Regress on the instrument and controls ; obtain (prediction).
  2. Regress on the predicted and controls .

• Special case: one instrument and no controls:

• Researchers may have multiple instruments or consider transformations of the initial instrument.

• If the exogeneity condition holds for , then the conditional moment implies unconditional moments:

  for any vector of instruments with .

• This leads to the question: how to choose to minimize the asymptotic variance of the GMM estimator of ?

• Choice of does not affect identification of the causal effect, but it affects the precision of the 2SLS (or GMM) estimator.

• We overview classical results on optimal instruments.

• For simplicity, restrict to conditional homoscedasticity:

• Setting and Goal

  • Assume only is endogenous and scalar; denote .
  • Consider moment conditions for GMM based on instruments with :

  • Define .

• GMM Estimator

  • The GMM estimator satisfies
    and

• Asymptotics
  • Asymptotic normality:
    where

  • The optimal minimizes the asymptotic variance in .

• Theorem (First-order condition at the optimum; cf. Newey & McFadden, 1994, §5.4; Chamberlain, 1987):
  If an efficient instrument exists, it satisfies
  i.e. holds at the efficient choice, not for arbitrary instruments .

• By iterated expectations and the condition , the optimality condition reads:

• Under conditional homoscedasticity

  the condition is satisfied by

• Since multiplying by a constant matrix does not change efficiency, minimizes asymptotic variance.
• The resulting optimal-object expression (semi-parametric efficiency) is:

• Interpretation:
  • The optimal instrument is the regression function .
  • attains the semi-parametric efficiency bound (see Van der Vaart, 1998, Ch.25).

• may be high-dimensional and challenging to estimate nonparametrically.
• With few instruments, Newey & McFadden (1994) suggest series (sieve) estimators for the regression .
• The optimal instrument improves precision but does not affect identification.

• Key Terms: Causal inference, RCT, treatment effects, average treatment effect (ATE), conditional independence, propensity score, IV methods, local average treatment effect (LATE).
• References:
  • Angrist & Pischke (2009)
  • Imbens & Rubin (2015)

• DID
• PSM
• RDD

• Definition:

  • Difference-in-Differences (DID) is a statistical technique used to estimate causal effects by comparing the outcomes of treatment and control groups before and after an intervention.

• Objective:

  • To isolate the effect of a specific treatment or policy by controlling for unobserved factors that could influence the outcome.

• Applicable Issues:

  • Policy analysis, economic interventions, program evaluations in finance, and social sciences.

• Model / Formula:

• Assumptions:

  • Parallel Trends Assumption: The treatment and control groups would have followed the same trajectory in the absence of treatment.

• Causal Inference Analysis:

  • The DID approach helps mitigate bias from confounding factors by ensuring that any changes in outcomes can be attributed to the intervention rather than other influences.

• Treatment Group:

  • The group that receives the intervention.

• Control Group:

  • The group that does not receive the intervention, used for comparison.

• Time Periods:

  • Pre-treatment: Measurements taken before the intervention.
  • Post-treatment: Measurements taken after the intervention.

• Context:

  • Analyzing the effect of a financial deregulation (e.g., the repeal of Glass-Steagall Act) on bank performance.

• Data:

  • Treatment Group: Banks affected by deregulation.
  • Control Group: Similar banks that weren’t affected.

• Observation:

  • Compare key financial metrics (e.g., ROA, stock performance) before and after deregulation to estimate its impact.

1. Identify treatment and control groups.
2. Collect longitudinal data for both groups before and after the intervention.
3. Calculate changes in outcomes for both groups.
4. Apply the DID formula to estimate the causal effect.
5. Validate the parallel trends assumption using pre-treatment data.

• Assumption Validity:

  • The parallel trends assumption may not be justified in all cases.

• Confounding Variables:

  • External factors can skew results; controlling for these factors is crucial.

• Data Quality:

  • Requires reliable longitudinal data and appropriate sample sizes for meaningful results.

• Latest Literature:
  • Machine learning techniques can be applied to enhance DID models by:
    • Improving Propensity Score Matching: Using algorithms to better pair treatment and control groups.
    • Feature Selection: Identifying key variables that affect treatment outcomes and adjusting for them.
    • Robustness Checks: Applying machine learning methods to assess the validity of findings from traditional DID.

• Summary:

  • DID is a powerful tool for estimating causal effects in the absence of randomization, particularly useful in the fields of economics and finance.

• References:

  1. Ashenfelter, O. (1978). "Estimating the Effect of Training Programs on Earnings." Review of Economics and Statistics, 60(1): 47–57.
  2. Card, D. (1990). "The impact of the Mariel boatlift on the Miami labor market."
  3. Goodman-Bacon, A. (2018). "Difference-in-Differences with Variation in Treatment Timing."
  4. Doudchenko, N., & Imbens, G. W. (2016). "Balancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis." NBER Working Paper 22791.

• Definition:

  • Propensity Score Matching (PSM) is a statistical technique used to estimate the effect of a treatment or intervention by accounting for the covariates that predict receiving the treatment.

• Objective:

  • To reduce selection bias by matching treated and control units with similar characteristics based on their propensity scores.

• Applicable Issues:

  • Policy evaluations, treatment effects in healthcare, finance, and social sciences.

• Model / Formula:

  • Propensity Score:

    • where indicates treatment and includes covariates.

• Assumptions:

  • Ignorability: The potential outcomes are independent of treatment assignment given the observed covariates.
  • Common Support: There is overlap in the distribution of covariates between treated and control groups.

• Causal Inference Analysis:

  • PSM aims to create a quasi-experimental condition that allows the estimation of causal effects without randomization by balancing covariates.

• Treatment Group:

  • Individuals receiving the intervention or treatment.

• Control Group:

  • Individuals not receiving the intervention, used for comparison.

• Propensity Score:

  • The probability of receiving the treatment given a set of observed covariates.

• Context:

  • Evaluating the impact of financial literacy training on savings behavior.

• Data:

  • Treatment Group: Individuals who completed the training.
  • Control Group: Similar individuals who did not participate.

• Process:

  • Match participants based on their propensity scores to estimate the training's effect on savings rates.

1. Estimate the propensity score using a logistic regression model (or other methods) to predict treatment assignment based on covariates.
2. Match treated and control units based on their propensity scores (e.g., nearest neighbor, caliper matching).
3. Assess the balance of covariates post-matching to ensure similarity between groups.
4. Estimate treatment effects using the matched sample.

• Assumption Validity:

  • The method relies heavily on the ignorability assumption; unmeasured confounders can bias results.

• Matching Quality:

  • Poor matching can lead to inadequate control of bias; it’s essential to check balance.

• Data Quality:

  • Requires a rich dataset with sufficient covariates to ensure valid estimates.

• Latest Literature:
  • Machine learning techniques are being increasingly integrated into PSM to enhance model estimation:
    • Propensity Score Estimation: Use of advanced algorithms (e.g., Random Forest, Gradient Boosting) to improve the accuracy of propensity score estimation.
    • Automation of Matching Process: ML algorithms can automate the matching of treated and control groups more effectively than traditional methods.
    • Robustness and Generalization: Assessing treatment effects across various populations by leveraging machine learning’s predictive capabilities.

• Summary:

  • PSM is a valuable tool for estimating causal effects in observational studies, particularly useful in finance and economics.

• References:

  1. Rosenbaum, P. R., & Rubin, D. B. (1983). "The central role of the propensity score in observational studies for causal effects."
  2. Imbens, G. W. (2004). "Nonparametric estimation of average treatment effects under exogeneity: A review."
  3. Lechner, M. (2002). "Some practical issues in the evaluation of labor market policies."
  4. Leuven, E., & Sianesi, B. (2003). "PSMATCH2: Stata Module to Perform Full Mahalanobis and Genetic Matching."

• Definition:

  • Regression Discontinuity Design (RDD) is a quasi-experimental pretest-posttest design that capitalizes on a cutoff or threshold to assign treatment, allowing for causal inferences about the effect of an intervention.

• Objective:

  • To estimate causal effects by exploiting a discontinuity in the assignment of treatment based on an observed variable.

• Applicable Issues:

  • Policy evaluations, program assessments in education and health, and financial interventions.

• Model / Formula:

  • The basic RDD model can be expressed as:
    • Where is the outcome variable, is an indicator for whether the treatment is applied, and captures the functional form of the covariate around the cutoff.

• Assumptions:

  • Continuity Assumption: The potential outcomes are continuous at the cutoff; no abrupt changes other than treatment effect.
  • Local Randomization: Units near the cutoff are considered as if randomly assigned between treatment and control groups.

• Causal Inference Analysis:

  • RDD allows for the estimation of causal effects by comparing outcomes just above and just below the cutoff, assuming that all other factors remain constant.

• Running Variable:

  • The continuous variable that determines treatment assignment based on a specified cutoff.

• Cutoff Point:

  • The threshold at which the treatment changes (e.g., income level, test score).

• Treatment Group:

  • Units just above the cutoff that receive the treatment.

• Control Group:

  • Units just below the cutoff that do not receive the treatment.

• Context:

  • Evaluating the impact of a minimum wage increase on employment levels.

• Data:

  • Running Variable: Average wage level of firms.
  • Cutoff Point: Minimum wage legislation threshold.

• Observation:

  • Compare employment trends for firms just above and just below the minimum wage threshold to estimate the policy's effect.

1. Identify the running variable and the cutoff point for treatment assignment.
2. Collect data on the outcome variable and covariates around the cutoff.
3. Perform graphical analysis to visualize the discontinuity at the cutoff.
4. Estimate the causal effects using local polynomial regression or parametric methods.
5. Check the robustness of results through sensitivity analyses and various bandwidths.

• Assumption Validity:

  • Results are only valid under the assumption that no other factors change at the cutoff.

• Limited Generalization:

  • The results may only apply to units near the cutoff; the external validity can be limited.

• Data Quality:

  • Requires large datasets with precise measures around the cutoff to ensure reliable estimates.

• Latest Literature:
  • Recent studies have begun integrating machine learning techniques into RDD to enhance causal inference:
    • Flexible Functional Forms: Machine learning algorithms (e.g., Random Forests, Neural Networks) can model nonlinear relationships around the cutoff.
    • Automated Bandwidth Selection: Using ML approaches to determine optimal bandwidth can improve estimation accuracy.
    • Robustness Checks: ML methods can help validate RDD findings by comparing predictions from different models.

• Summary:

  • RDD is a robust method for estimating causal effects where random assignment is not feasible, particularly relevant in economics and finance.

• References:

  1. Imbens, G. W., & Lemieux, T. (2008). "Regression Discontinuity Designs: A Guide to Practice."
  2. Lee, D. S., & Lemieux, T. (2010). "Regression Discontinuity Designs in Economics."
  3. Cattaneo, M. D., Idrobo, N., & Titiunik, R. (2019). "A Practical Introduction to Regression Discontinuity Designs."
  4. Gelman, A., & Imbens, G. W. (2019). "Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs."

• Importance: Model selection and sparsity in high-dimensional data.
• Focus:
  • Post-selection inference
  • Lasso estimator as a variable selection tool
  • Double selection method
• Related Chapters:
  • Chapter 5: Post-machine learning inference
  • Chapter 6: Instrumental variable models
  • Chapter 7: Further developments

• Two-step inference:
  • First, select a model.
  • Then, report results as if true.
• Challenge: Ignoring the selection step can lead to misleading results.
• Context: Based on the works of Leeb and Pötscher (2005).

• Assumption: Sparse Gaussian linear model
  • is non-singular.
• Models:
  • : Sparse (R) vs. Unrestricted (U)

• Decision rule for inclusion of :

• Asymptotic properties (standard result; see Gaillac & L'Hour 2025, Ch. 4):
  • Consistency of model selection as .

• Post-selection estimator given by:

• Distribution results indicate deviation from Gaussian even under selection.

• Lasso estimator defined as:
  • Balances fit (MSE) and sparsity (L1 norm).
• Assumptions: Sparse Gaussian linear model, with sparsity constraint on .

Theorem: Consistency of Lasso

• The Lasso estimator is consistent in :

• Implications of penalty strength and eigenvalue conditions.

• Regularization Bias: Bias from including/removing variables leading to erroneous inference.
• Risk of dual objectives: (selecting variables and accurate estimation).

1. Selection on Treatment: Regress on using Lasso.
2. Selection on Outcome: Regress on using Lasso.
3. Final Estimation: OLS regression of on selected and .

• Outcome:
  • Provides valid inference on treatment effects.

• Data: French Enquête Emploi 2017–2019, 162,254 observations (Gaillac & L'Hour 2025, §4.6; reproduces L'Hour 2020).
• Modeling: Log of monthly salary explained by education levels and other controls.
• Importance:
  • Double selection method improves precision in estimating treatment effects.

• Key Concepts:
  • Post-selection inference, Lasso estimator, regularization bias, double selection method.
• Recommendations:
  • Understanding the nuances of variable selection and estimation under uncertainty.

• Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. JRSS-B, 58(1): 267–288.
• Belloni et al. (2014) - Accessible econometric references.
• R Packages: hdm for practical implementation.

• Focus: Importance of generalization in econometric modeling.
• Goals:
  • Understand methodologies for assessing model performance.
  • Explore overfitting and its implications.
• Contents:
  • Statistical learning theory
  • Cross-validation techniques
  • Regularization methods

• Definition: Generalization refers to a model's ability to perform well on unseen data.
• Key Concept: Bias-variance tradeoff impacts prediction accuracy:
  • Bias: Error due to approximating complex truths.
  • Variance: Error due to model sensitivity to fluctuations in training set.

• Framework: Analyzes relationships between generalization abilities of learning algorithms and model complexity.
• Risk Function:
  • : Loss function.
  • : Expected risk (average loss).

• Definition: Complexity measures the capacity of a model to fit data.
• Types:
  • Flexible models: Can fit diverse patterns (e.g., high-degree polynomials).
  • Rigid models: Offer less variance (e.g., linear models).
• Implication: Increased complexity can lead to overfitting.

• Definition: Occurs when a model captures noise in training data rather than underlying patterns.
• Indicators:
  • High accuracy on training set vs. low accuracy on validation/test set.
• Consequences: Leads to poor generalization and predictive performance.

• Purpose: Assess model performance and avoid overfitting.
• Methods:
  • K-fold Cross-Validation: Divide data into subsets; train on and validate on the remaining.
  • Holdout Method: Split dataset into training and test sets.
• Goal: Ensure the model generalizes well to new data.

• Purpose: Combat overfitting by adding a penalty for complexity.
• Common Methods:
  • Lasso Regression:

  • Ridge Regression:

• Effect: Reduces model complexity and enhances prediction stability.

• Common Criteria:
  • AIC (Akaike Information Criterion):
  • BIC (Bayesian Information Criterion):

• Use: Select models based on the trade-off between goodness of fit and model complexity.