An open course in econometrics · Ian Helfrich
The Econometric
Argument
An introduction to econometrics: probability, regression, causal identification, and decisions under uncertainty.
Chapters 1 and 2 require fractions and basic algebra; chapter 1 also previews convergence in probability, which chapter 4 develops. Later chapters add derivatives, matrix notation, and rate notation. Each chapter lists its prerequisites before it begins and names the earlier chapters it relies on.
12 compact chapters · 3 experiments · worked problems
Development edition, September 2026
Observational equivalence
160 synthetic observations · seed 2026 · standardized units
A one-unit intervention raises the mean outcome by 0.50. The observational distribution has not changed.
Sample OLS slope: 0.843. Sampling noise explains its distance from 1; it cannot select the causal world.
Why the data cannot choose →Inspect the model and its limits
Let U and V be independent standard normal variables. Set D = U and Y = bD + (1 − b)U + V. Observationally, Y = D + V for every b. Under do(D = d), U is unchanged and the mean is bd. The orange line is a model implication, never an estimate from these dots.
The slider range [−1, 2] is chosen for display. The observed distribution does not restrict b to that range. U is an unobserved cause; its equality with observed D in this deliberately stark construction makes conditional causal overlap fail.
Each unit's observed values remain fixed as b changes. No animation, fitting algorithm, or larger sample can identify b from this observational law alone.
What did the number leave out?
Compare two service designs: one offers faster appointments, the other gives participants more control. The ranking depends on how both outcomes are measured and weighted.
Open the measurement investigationDoes training improve productivity?
Workers who take more training produce more. Training may improve productivity, or more productive workers may be more likely to enroll.
The chapters use this example to study selection, estimation, and the evidence needed to justify expanding a training program.
- TargetWhat would the answer mean?
- ObserveWho enters the data?
- IdentifyWhich rival worlds remain?
- EstimateWhat does the procedure recover?
- QuantifyHow could this answer fail?
- DecideWhat action survives the uncertainty?
The book
Contents
Reading progress is optional and stays on this device.
Open the printable edition →The chapters are ordered by prerequisite. Use the filters for shorter reading lists.
- 01Estimands and identification
Separate the question a dataset can answer from the question a decision requires.
Estimands · Identification · Causal models
Frame · about 25 min · ready to read - 02Conditional expectation and composition
Use conditioning and expectation to diagnose changes in composition before interpreting an average.
Probability · Conditional expectation · Composition
Frame · about 25 min · ready to read - 03Least squares and projection
Derive least squares, interpret residualization, and distinguish a projection coefficient from a causal effect.
OLS · Projection · FWL · Omitted variables
Estimate · about 30 min · ready to read - 04Sampling uncertainty and standard errors
Distinguish sampling uncertainty from bias and choose an uncertainty calculation that matches the design.
Inference · CLT · Sandwich variance · Clustering
Estimate · about 30 min · ready to read - 05Randomized experiments
Construct a randomized comparison, distinguish its estimand from uptake effects, and inspect the design threats.
Experiments · Randomization · Potential outcomes
Identify · about 30 min · ready to read - 06Measurement error, missing data, and selection
Trace measurement, missingness, and conditioning decisions into the estimand and its identification.
Measurement error · Missingness · Selection · DAGs
Identify · about 30 min · ready to read - 07Instrumental variables and the LATE
Derive an instrumental-variables target, distinguish validity from strength, and examine what happens when exclusion fails.
IV · LATE · Weak instruments · GMM
Identify · about 30 min · ready to read - 08Panel data and difference-in-differences
Derive difference-in-differences, separate fixed effects from identification, and quantify a transparent trend sensitivity.
Panel data · Fixed effects · DiD · Sensitivity
Identify · about 30 min · ready to read - 09Regression discontinuity and synthetic control
Explain what a cutoff or a synthetic comparison identifies, including the conditions that make a convincing picture insufficient.
Regression discontinuity · RD · Synthetic control
Identify · about 30 min · ready to read - 10Double machine learning
Derive an orthogonal partial-linear score and understand the role and limits of sample splitting and nuisance learning.
Machine learning · DML · Cross-fitting · Orthogonality
Generalize · about 35 min · ready to read - 11Likelihood, forecasting, and transportability
Connect likelihood, dependent-data forecasting, predictive coverage, and the assumptions needed to transfer an effect.
Likelihood · GLM · Time series · Conformal · Transport
Generalize · about 35 min · ready to read - 12Decisions under partial identification
Translate effects into a constrained decision, distinguish sensitivity from sampling uncertainty, and assemble an auditable empirical argument.
Decision theory · Policy learning · Bounds · Interference
Generalize · about 30 min · ready to read
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For classrooms and working analysts
Teaching materials
The teaching guide includes a twelve-week schedule, a ninety-minute workshop, and a capstone on a training-program decision.
Teaching plans, workshop, and capstone →
Arrange one-to-one tutoring, starting with a 20-minute intro call →
Readings
References include Hansen’s Econometrics and papers on difference-in-differences, instrumental variables, machine learning, and policy evaluation.
The working handbook Econometrics Across Disciplines takes up identification, measurement, and the timing of information in three chapters, with complete proofs and independent Python, Julia, and R calculations.
About this edition
The advanced chapters introduce topics that need further reading and problem sets for graduate study. This is a development edition.
Edition notes →