Lesson 14

Known-Form Heteroskedasticity

Big question

What if the shape of the variance function is known up to scale?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain known-form heteroskedasticity in plain language.
  • Use weighted least squares correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

Heteroskedasticity is about changing uncertainty. In known-form heteroskedasticity, students focus on known-form WLS transformation and learn how to describe that pattern without confusing it with coefficient bias.

Key terms

Weighted least squares
A least-squares method that gives lower weight to observations with higher error variance.
Linear probability model
An OLS model for a binary outcome whose error variance depends on the fitted probability.
Robust standard error
A standard error designed to remain asymptotically valid under heteroskedasticity.

Core formula

Var(ui∣xi)=sigma2hi;dividebysqrt(hi)Var(u_i|x_i)=sigma^2 h_i; divide by sqrt(h_i)

Use plain-language interpretation before algebra.

Example

MODULE8_INCOME_SAVINGS_SYNTHETIC supports original Ceteris Lab practice for known-form heteroskedasticity. Synthetic files are clearly labeled as synthetic, and installed course datasets are used only when the file is present.

Interactive visual

WLSTransformationExplorer: adjust the controls, classify the diagnostic evidence, and write one sentence explaining the implication for inference.

Original Module 8 visual for Known-Form Heteroskedasticity.

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

Known-Form Heteroskedasticity Python example

Known-Form Heteroskedasticity 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 4Load the dataset into a pandas DataFrame.
  4. Line 5Create or update a Python object used in the analysis.
  5. Line 6Create or update a Python object used in the analysis.
  6. Line 7Add an intercept column to the regression design matrix.
  7. Line 8Estimate an ordinary least squares regression.
  8. Line 9Create or update a Python object used in the analysis.
  9. Line 10Display a result so students can inspect the output.

Python walkthrough

  1. 1Load the Python packages needed for data, regression, diagnostics, or plotting.
  2. 2Read an installed Ceteris Lab dataset from a browser-safe public path.
  3. 3Estimate the baseline model before changing the covariance method or weights.
  4. 4Print diagnostic evidence or a coefficient comparison so students can inspect the result.
  5. 5Interpret the output as practice evidence and avoid making real empirical claims from synthetic data.

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

MODULE8_INCOME_SAVINGS_SYNTHETIC

Estimated time

25 to 40 min

Packages

pandas, numpy, statsmodels, patsy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect MODULE8_INCOME_SAVINGS_SYNTHETIC.
  • Run the Python cells connected to Weighted Least Squares Intuition.
  • Interpret the output using heteroskedasticity and robust standard errors.

Common errors

  • File not found: check that module8_income_savings_heteroskedastic.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-8/module8_income_savings_heteroskedastic.csv")
df.head()

Interactive activity

WLS Transformation Explorer

Known-Form Heteroskedasticity

change the inputs, inspect the feedback, and decide whether robust inference, diagnostics, WLS, or reporting caution is needed.

MODULE8_INCOME_SAVINGS_SYNTHETIC

Inputs

Visual preview

Changing variance across fitted values

fitted valueresidual spread

A wider fan means the uncertainty changes across observations. Robust standard errors adjust inference; WLS needs a defensible variance model.

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 should a careful Module 8 report include for Known-Form Heteroskedasticity?

Quick quiz

What should a careful Module 8 report include for Known-Form Heteroskedasticity?

Quick quiz

Why is SLEEP75 a reasonable practice dataset here?

Key takeaway

Known-Form Heteroskedasticity helps students diagnose changing variance and choose inference or weighting methods without overclaiming what those methods can fix.

Download dataDownload resultsDownload interpretation