Lesson 6

Inconsistency and Asymptotic Bias

Big question

Why can more data make a bad estimator confidently wrong?

Lesson progress

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain inconsistency and asymptotic bias in plain language.
  • Use plim correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

If a regressor is correlated with the error, OLS converges to the wrong number. The gap is asymptotic bias.

Key terms

plim
A shorthand for probability limit.
omitted-variable inconsistency
Large-sample bias caused by leaving out a relevant correlated variable.
skewness
A statistic measuring distribution asymmetry.

Core formula

plim(betahat1)−beta1=Cov(x1,u)/Var(x1)plim(beta_hat_1) - beta_1 = Cov(x1,u) / Var(x1)

Use plain-language interpretation before algebra.

Example

Inconsistency and Asymptotic Bias uses simulated data so students can see the large-sample mechanism without inventing empirical results.

Interactive visual

InconsistencyDirectionSimulator

Original Module 5 visual for Inconsistency and Asymptotic Bias.

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

Inconsistency and Asymptotic Bias Python example

Inconsistency and Asymptotic Bias 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 10Create or update a Python object used in the analysis.
  10. Line 11Run this Python instruction as part of the lesson workflow.
  11. Line 12Create or update a Python object used in the analysis.
  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 Inconsistency and Asymptotic Bias.
  • Interpret the output using inconsistency and endogeneity.

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

InconsistencyDirectionSimulator

Set the direction of asymptotic bias

Move the covariance and variance inputs to see the sign of the large-sample bias.

Simulation

Inputs

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

What is asymptotic bias?

Quick quiz

Which reporting habit is most important in Inconsistency and Asymptotic Bias?

Quick quiz

Why is 401KSUBS a reasonable practice dataset here?

Key takeaway

Inconsistency and Asymptotic Bias helps students separate large-sample approximation from valid research design.

Download dataDownload resultsDownload interpretation