Causal Academy/Causal Inference with Linear Regression: A Modern Approach (Part I)

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Causal Inference with Linear Regression: A Modern Approach (Part I)

  • Course
  • 97 Lessons

Learn how linear regression can be used as an estimation tool in causal inference. Understand what regression coefficients represent, under which assumptions they can be interpreted as causal effects, how violations of unconfoundedness affect results, and how to use robustness tests and sensitivity analysis to assess and cope with these violations.

Watch a short overview of the course and what you’ll learn.

About this course

This 10-hour course is the first part of a series on causal inference with linear regression. It is designed for people who want to understand precisely how linear regression can be used as an estimation tool in causal analysis, and under which conditions regression coefficients can be interpreted as causal effects.

The course starts with a concise introduction to the core concepts from causal inference that are needed for the remainder of the course. This includes causal quantities, conditional exchangeability, and the adjustment formula.

From there, the course turns to the mechanics of linear regression, with an emphasis on how regression coefficients are computed and how linear regression can be understood as an approximation to the conditional expectation function.

Building on this, the course develops the connection between causal quantities and linear regression as a statistical tool. Linear structural causal models are introduced to define the causal parameters of interest, and their relationship to regression equations is made explicit. This allows us to study the exact conditions under which regression coefficients recover these causal parameters, and when they do not.

Finally, the course covers robustness tests and sensitivity analysis for unobserved confounding. The focus here is on how violations of assumptions such as unconfoundedness affect OLS-based causal estimates, how this sensitivity can be quantified, and how these methods can be implemented in Python.

What makes this course different

A clear mapping between causal inference and linear regression

The course makes explicit how linear regression relates to causal inference. Rather than relying on abstract assumptions such as exogeneity or ambiguous arguments about unbiasedness for “true parameters,” the connection between regression coefficients and causal quantities is developed in a transparent and structured way.

A strong focus on robustness and sensitivity analysis

A central part of the course is dealing with violations of the unconfoundedness assumption. In particular, the course covers robustness tests and the sensitivity analysis framework of Cinelli & Hazlett (2020), which allows you to quantify how strong unobserved confounding would need to be to change your conclusions. These tools are essential in practice, but rarely covered in detail.

Grounded in modern causal inference literature

The material is based on high-quality sources in causal inference, including work by Judea Pearl, Joshua Angrist & Jörn-Steffen Pischke, Carlos Cinelli, and Chad Hazlett.

Extensive use of coding demonstrations

The course makes heavy use of coding demonstrations and exercises. The goal is to make sure that everything we discuss conceptually also becomes tangible and applicable in practice.

What this course covers

The course develops a structured understanding of how linear regression can be used for causal inference.

Causal foundations

  • Individual, conditional, and average treatment effects (ITE, CATE, ATE) for both categorical and continuous treatments

  • Conditional exchangeability (unconfoundedness)

  • The adjustment formula

  • Directed acyclic graphs (DAGs)

  • The role of bidirected (double-headed) arrows

  • The backdoor criterion

Linear regression and the CEF

  • How OLS determines regression coefficients

  • The Frisch–Waugh–Lovell theorem

  • The relationship between linear regression and the conditional expectation function

The relationship between causal inference and OLS

  • Linear structural causal models

  • The meaning of structural causal parameters

  • The relationship between structural causal parameters and regression coefficients

  • The conditions under which OLS coefficients recover causal parameters

Violations, robustness, and sensitivity analysis

  • The consequences of violations of unconfoundedness for OLS-based causal estimates

  • When and how robustness tests can be used to stress-test causal assumptions

  • Sensitivity analysis for unobserved confounding based on the framework of Cinelli & Hazlett (2020), including interpretation of sensitivity measures and practical implementation in Python

Who this course is for

This course is intended for data scientists, analysts, and other quantitative practitioners who want a deep understanding of how and when linear regression can be used for causal inference.

A basic understanding of probability and statistics is assumed. Some familiarity with Python is helpful, as coding examples and exercises will be in python.

Upon completion of the course, you will receive a certificate of completion from Causal Academy.

Next step

This course develops the foundation for using linear regression in causal analysis by making the connection between regression and causal inference explicit.

In Part II of the course series, this foundation is extended to more flexible linear modelling approaches, including feature engineering and double machine learning based on LASSO and OLS, and embedded in a complete end-to-end applied causal analysis workflow.

Contents

A Crash Course on The Basics of Causal Inference

Introduction
Slides
How to Use This Course
Course Guidelines
What is Causal Inference and why do we need it?
Individual Treatment Efffect
(Conditional) Average Treatment Effect
The do-operator
Ignorability
Conditional Ignorability & The Adjustment Formula
Causal Graphs
Graph Patterns
Blocking Paths
Backdoor Adjustment
Double-headed arrows & The do-operator in Causal Graphs
Coding Example: Generating Data
Coding Example: Observational & Experimental study (no adjustment)
Coding Example: Observational study (with adjustment)
Extending to Continuous Treatments
Recap
Theoretical Questions
Coding Exercise
Resources

The Mechanics of Linear Regression

Introduction
Slides
Prerequisites
Linear Regression
Closed-Form Solution
OLS Residuals
From Population to Sample
The CEF
CEF Properties
Linear Regression as CEF Approximator
Closed-Form Formula Revisited
FWL & The Regression Anatomy Formula
Changing Slope Coefficients Part I
Changing Slope Coefficients Part II
Recap
Theoretical Questions
Coding Exercise
Resources

When Linear Regression Coefficients Reflect Causal Effects

Introduction
Slides
Recap Causality Basics
Total, Direct & Indirect Effects
Structural Causal Models
Linear SCMs & The do-operator
The Meaning of Structural Parameters Part I
The Meaning of Structural Parameters Part II
Linear Structural Equation vs Linear Regression Equation
Identifying Structural Coefficients for All Variables Part I
Identifying Structural Coefficients for All Variables Part II
Coding Example: Independent Structural Errors
Coding Example: Dependent Structural Errors
Linear Regression: From Searching for The ‘"True Model’ to Control Tool
The Single-Door Criterion
Coding Example: Single-Door Criterion
Total Effects In Terms of Structural Parameters
The Back-Door Criterion
Coding Example: Back-Door Criterion
Helping Out The Fitness Team
On The Nuisance of Control Variables
Econometrics & Causality
Recap
Theoretical Questions
DAGitty
Coding Exercise
Resources

Robustness Tests & Sensitivity Analysis

Introduction
Slides
What Are Robustness Tests?
Detecting Omitted Variable Bias Part I
Coding Example
Detecting Omitted Variable Bias Part II
Coding Example
What Is Sensitivity Analysis?
The Real-World Example
The Traditional OVB Framework
Applying The Traditional OVB Formula
(Partial) R-Squared
A Partial R-Squared Reparametrization of the OVB Formula
Applying The Reparametrized OVB Formula
Bounding The Strength of An Omitted Confounder
Applying The Bounding Procedure
Extreme Scenario Analysis
Applying Extreme Scenario Analysis
Summary Metrics
Applying Summary Metrics
Extending With Inference & Multiple Omitted Confounders
The Package: PySensemakr
Recap
Theoretical Questions
Coding Exercise
Resources

Part 2 & Recommendations

Part 2 & Recommendations
Reading List & Sources
Feedback and Reviews

Reviews

Josep M.

Managing Director @ Accenture

"A very good choice. Step by step it builds confidence and helps differentiate the subtleties of the argument."

Akarapat C.

Data Expert @ Athentic Consulting

“High quality (both presentation slides and narration). This is the course I am looking for to learn causal inference.”

⭐ 4.9/5 average rating

Based on 30+ reviews from previous versions of this course on Udemy.