Causal Inference with Linear Regression: A Modern Approach

Explore how linear regression models determine slope coefficients and interpret causal effects under multiple variables, and connect linear regression to the conditional expectation function for future causal inference.

Explore the distinction between population and sample level estimation, and learn core tools—linearity of the expectation, law of iterated expectation, and derivatives—to prepare for linear regression analysis.

Explore linear regression as a model linking a dependent variable to multiple independent variables, with an intercept and beta coefficients, noting associations rather than causality and residuals as unexplained variation.

Compute linear regression coefficients using ordinary least squares by minimizing the expected squared residual, yielding the closed-form beta = (E[xx^T])^{-1} E[xy].

Learn how ordinary least squares residuals are uncorrelated with regressors by showing E[x epsilon]=0 and E[epsilon]=0 in y equals x transpose beta_ols plus epsilon, with x including an intercept.

From population to sample, this lecture applies ordinary least squares to relate height and weight, highlighting sample averages, residuals, and the conditional expectation function with real data.

The conditional expectation function (CEF) describes the expected value of y given x1 through xk and summarizes how y varies with education by the mean at each education level.

Understand how the conditional expectation function decomposes Y into E[Y|X] and a mean-independent residual, with the residual uncorrelated to any function of X. Learn why the CDF minimizes squared error.

See how linear regression serves as the best linear approximation to the conditional expectation function, exact in population when CF is linear, or closest via OLS when not.

Apply the Frisch-Waugh-Lovell theorem to isolate x1's effect by regressing out other variables and comparing residuals, clarifying that regression coefficients measure association, not causation.

Explore how slope coefficients in linear regression change when adding variables like weight loss goals or calorie intake, and how holding other factors constant reshapes interpretation of exercise effects.

explain how slope coefficients in OLS capture the linear association while holding others constant. show how adding weight loss goals and calorie intake reshapes coefficients and informs causal inference.

Explore linear regression mechanics, including ordinary least squares, slope coefficients, and conditional associations, and learn when regression coefficients can reflect causal relationships.

Examine when linear regression coefficients reflect causal effects, and how to connect the conditional expectation function to interventional expectations, including total versus direct effects and linear structural causal models.

We recap the core causal concepts from module one, including the average treatment effect, conditional average treatment effect, and individual treatment effect, and how the do operator affects estimation.

Explore total, direct, and indirect causal effects in linear models, illustrating how T affects Y via direct and indirect paths using the do operator.

Explore how structural causal models assign endogenous variables via structural functions from parents and exogenous factors, and how these assignments form directional causal graphs.

Learn linear structural causal models, with endogenous and exogenous variables, and apply the do-operator to understand interventions, observational and post-intervention distributions.

Interpret theta_ik as the direct effect of a one unit increase in vk on VI in a linear system, via the controlled direct effect, independent of other variable values.

The lecture explains that each theta_ik is the direct effect of a unit increase in v_k on v_i under intervention. The slope is constant, showing effect homogeneity.

Explore how linear structural equations define direct causal effects with theta parameters, contrast with regression coefficients that measure associations, and learn when regression can reflect causal effects.

Explore when linear regression coefficients reflect true causal effects, focusing on direct effects, exogeneity, and the conditions for r1 to equal theta1 in simple and multi-variable settings.

Show when the population regression of y on x1 and x2 recovers theta1 and theta2, given ui independence from x1 given x2 or ui independence from x2 given x1.

Run linear regression of y on x1, x2, and x3 to estimate direct effects. With independent errors and conditioning on x2 and x3, coefficients closely match true direct effects.

Explore how dependent structural errors bias regression coefficients in a linear model of y on x1, x2, x3, and identify when some coefficients remain unbiased.

Reframe linear regression as a control tool for causal effects by guiding the selection of a z set to de-bias the treatment coefficient and reveal direct and total effects.

Apply the single-door criterion to identify the direct effect of t on y by blocking all paths except the direct one and avoiding descendants of y.

Apply the single door criterion to a causal graph to identify an adjustment set and estimate the direct effect of T on Y with linear regression, ensuring unbiased coefficients.

Explain how total effects combine direct and indirect pathways in linear structural models, express them via path coefficients, and apply the average treatment effect with do-interventions and back door criterion.

Explore the backdoor criterion to identify total and direct effects in linear regression, using a control set z to block noncausal paths while preserving causal ones.

Apply the backdoor criterion to identify valid adjustment sets, such as x3 or x4, that block backdoor paths and enable unbiased estimation of the total effect via regression.

Investigate how linear regression relates to causality, exogeneity, and structural equations, and learn Pearl's modern framework for causal inference to clarify econometrics' ambiguities.

Explore the DAGitty tool for creating, editing, and analyzing causal graphs, identify backdoor and single-door admissible sets, and learn to adjust for biasing paths to estimate causal effects.

Explore robustness tests and sensitivity analysis to assess omitted variable bias in causal inference, using causal graphs, single and backdoor criteria, and Pearl, Haslett, and Cinelli insights.

Explore robustness tests in linear regression, including evaluating homoscedasticity, using the Y test, and diagnosing omitted variable bias. Apply single door and back door criteria to assess causal interpretation.

Test for omitted variable bias by comparing treatment coefficients across single door adjustment sets, using informative robustness tests to detect causal-graph misspecification and strengthen confidence in unbiased estimates.

Use the single door criterion and adjustment sets like x1 x4 to estimate the direct effect of t on y via regression, and robustness tests to detect misspecifications.

Examine how robustness tests with z2 detect omitted variable bias and possible bias amplification from unobserved confounding between t and y, using z1 and z2 in the causal graph.

Apply the single door adjustment to estimate the direct effect of t on y, using x1 (and optionally x2), then evaluate robustness and shift toward sensitivity analysis to assess misspecification.

Explore sensitivity analysis for causal inference in linear regression, based on Cinelli and Hazlett, assessing how unobserved confounders and omitted variables could alter estimates with Python and R tools.

Assess how being harmed in attacks shapes attitudes toward peace, using a regression with female and village controls, one-hot village dummies, and a Cinelli-Hazlett sensitivity framework.

Explore how omitting a confounder z biases the estimated treatment effect via the omitted variable bias formula using c and f, guided by the Frisch-Waugh-Lovell theorem and contour plots.

Learn Cinelli and Hazlett’s method to bound an unobserved confounder’s strength using partial R-squared, comparing it to observed variables via CT and k in linear regression.

Apply the bounding procedure to assess sensitivity to unobserved confounders using partial r squared, compare with female as a strong confounder, and introduce extreme scenario analysis.

Explore extreme scenario analysis in causal inference with linear regression, assessing how maximal confounder–outcome association and varying treatment–confounder r squared affect bias and the robustness of causal estimates.

Apply extreme scenario sensitivity analysis to gauge how unobserved confounders could explain away the treatment effect by varying the confounder’s partial R-squared with outcome and treatment.

Discover three metrics for sensitivity analysis in ols regression, including the robustness value and partial r-squared, to gauge confounding using contour plots.

Explore sensitivity analysis in causal inference by reporting partial R-squared and robustness value, examining confounders and treatment effects with Cinelli-Hazlett metrics.

Extend sensitivity analysis to t statistics, p values, and confidence intervals using contour plots to gauge significance under omitted confounders and multiple confounders, including robustness value RV1 alpha=0.05.

Apply the sense maker package to perform sensitivity analysis on a linear regression, assess omitted confounders, and interpret robustness values in a Darfur case study.

Explore robustness tests for omitted variables using back-door admissible additions and sensitivity analysis with partial R-squared and robustness value to assess confounding risk.

Part two expands causal inference with linear regression by embracing non-linear treatment effects and heterogeneity, refining bias handling, and guiding variable selection with recommendations and Pearl's reading list.