Lesson 15

Asymptotic Efficiency of OLS

Big question

Is every consistent estimator equally good?

Lesson progress

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain asymptotic efficiency of ols in plain language.
  • Use asymptotic variance correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

Two estimators can both converge to the right parameter, but one may have a tighter large-sample distribution.

Key terms

asymptotic variance
The large-sample variance scale of an estimator.
Lagrange multiplier statistic
A test statistic often computed as n times an auxiliary regression R-squared.
instrumental variables preview
A preview of using external variation to handle endogenous regressors.

Core formula

betatilde1=[sum(zi−zbar)yi]/[sum(zi−zbar)xi]beta_tilde_1 = [sum (z_i-zbar)y_i] / [sum (z_i-zbar)x_i]

Use plain-language interpretation before algebra.

Example

Asymptotic Efficiency of OLS uses simulated data so students can see the large-sample mechanism without inventing empirical results.

Interactive visual

AsymptoticEfficiencySimulator

Original Module 5 visual for Asymptotic Efficiency of OLS.

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

Asymptotic Efficiency of OLS Python example

Asymptotic Efficiency of OLS 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 4Load a Python library needed for data work or regression.
  5. Line 6Run this Python instruction as part of the lesson workflow.
  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 11Create or update a Python object used in the analysis.
  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 14Add an intercept column to the regression design matrix.
  14. Line 15Create a log version of the variable so coefficients can be read approximately as percentages.
  15. Line 16Create or update a Python object used in the analysis.
  16. Line 17Run this Python instruction as part of the lesson workflow.
  17. Line 18Create or update a Python object used in the analysis.
  18. Line 19Display a result so students can inspect the output.
  19. Line 20Create or update a Python object used in the analysis.
  20. Line 21Run this Python instruction as part of the lesson workflow.

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

35 to 55 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 Asymptotic Efficiency of OLS.
  • Interpret the output using asymptotic efficiency and simulation.

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

AsymptoticEfficiencySimulator

Compare estimator spread

Compare two consistent estimators and decide which is more efficient in large samples.

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 does asymptotic efficiency compare?

Quick quiz

Which reporting habit is most important in Asymptotic Efficiency of OLS?

Quick quiz

Why is BWGHT2 a reasonable practice dataset here?

Key takeaway

Asymptotic Efficiency of OLS helps students separate large-sample approximation from valid research design.

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