Lesson 7

Omitted Variable Inconsistency

Big question

Why does leaving out a relevant correlated variable not wash out?

Lesson progress

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain omitted variable inconsistency in plain language.
  • Use law of large numbers correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

Omitted-variable inconsistency is the population version of omitted-variable bias. It remains even with huge samples.

Key terms

law of large numbers
A result saying sample averages stabilize near population averages.
asymptotic normality
Approximate normality of an estimator in large samples.
histogram
A graph showing how observations are distributed across bins.

Core formula

plim(betatilde1)=beta1+beta2∗delta1;delta1=Cov(x1,x2)/Var(x1)plim(beta_tilde_1)=beta_1+beta_2*delta_1; delta_1=Cov(x1,x2)/Var(x1)

Use plain-language interpretation before algebra.

Example

Omitted Variable Inconsistency uses simulated data so students can see the large-sample mechanism without inventing empirical results.

Interactive visual

OmittedVariableInconsistencyMatrix

Original Module 5 visual for Omitted Variable Inconsistency.

wage_sample.csv

y variable

wage

The dependent variable. It is the outcome students want to explain.

x variable

education

The explanatory variable. It is used to describe changes in wage.

Live Python

Omitted Variable Inconsistency Python example

Omitted Variable Inconsistency Python example

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 3Load a Python library needed for data work or regression.
  4. Line 5Run this Python instruction as part of the lesson workflow.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Create or update a Python object used in the analysis.
  7. Line 8Create or update a Python object used in the analysis.
  8. Line 9Run this Python instruction as part of the lesson workflow.
  9. Line 10Run this Python instruction as part of the lesson workflow.
  10. Line 11Create or update a Python object used in the analysis.
  11. Line 12Run this Python instruction as part of the lesson workflow.
  12. Line 13Create or update a Python object used in the analysis.
  13. Line 14Create or update a Python object used in the analysis.
  14. Line 15Create or update a Python object used in the analysis.
  15. Line 16Add an intercept column to the regression design matrix.
  16. Line 17Run this Python instruction as part of the lesson workflow.
  17. Line 18Display a result so students can inspect the output.

Python walkthrough

  1. 1Load libraries and data or set a simulation seed.
  2. 2Build the model or simulation that matches the lesson question.
  3. 3Compute the statistic, graph, or summary table.
  4. 4Interpret the result as large-sample evidence, not automatic causality.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

SIMULATION

Estimated time

25 to 40 min

Packages

pandas, numpy, scipy

Expected output

Simulation output showing how estimates or test statistics behave as sample size changes.

Learning goals

  • Load and inspect SIMULATION.
  • Run the Python cells connected to Omitted Variable Inconsistency.
  • Interpret the output using omitted variables and probability limits.

Common errors

  • File not found: check that WAGE1.csv is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_csv("/data/module-5/WAGE1.csv")
df.head()

Interactive activity

OmittedVariableInconsistencyMatrix

Read the omitted-variable sign matrix

Combine the omitted effect sign and covariance sign to predict inconsistency.

Simulation

Inputs

Pick the next report ingredient

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

A regression coefficient describes a pattern unless the assumptions or research design support a causal interpretation.

Quick quiz

Why does omitted-variable inconsistency not disappear with more data?

Quick quiz

Which reporting habit is most important in Omitted Variable Inconsistency?

Quick quiz

Why is CRIME1 a reasonable practice dataset here?

Key takeaway

Omitted Variable Inconsistency helps students separate large-sample approximation from valid research design.

Download dataDownload resultsDownload interpretation