Lesson 16

Variance, Standard Errors, and Degrees of Freedom

Big question

What makes a coefficient estimate more or less precise?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain variance, standard errors, and degrees of freedom in plain language.
  • Use partial effect correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

Precision improves with more useful variation and larger samples, and worsens with larger error variance, stronger collinearity, and fewer residual degrees of freedom.

Key terms

partial effect
The predicted change in the outcome from changing one regressor while controls are fixed.
BLUE
Best Linear Unbiased Estimator under the Gauss-Markov assumptions.
variance inflation factor
A measure of how much collinearity inflates a coefficient variance.

Core formula

df=n−k−1df = n - k - 1

Use plain-language interpretation before algebra.

Example

The expression Var(beta_j_hat) = sigma squared over SST_j times one minus R_j squared organizes the main precision forces.

Interactive visual

Coefficient-precision simulator

Original Module 3 visual for Variance, Standard Errors, and Degrees of Freedom.

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

Variance, Standard Errors, and Degrees of Freedom Python example

Variance, Standard Errors, and Degrees of Freedom 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 5Load the dataset into a pandas DataFrame.
  5. Line 6Create a log version of the variable so coefficients can be read approximately as percentages.
  6. Line 7Add an intercept column to the regression design matrix.
  7. Line 8Create or update a Python object used in the analysis.
  8. Line 9Estimate an ordinary least squares regression.
  9. Line 10Display a result so students can inspect the output.

Python walkthrough

  1. 1Load the dataset and keep only variables needed for the current model.
  2. 2Construct the dependent variable and explanatory-variable matrix.
  3. 3Add a constant so the regression includes an intercept.
  4. 4Fit OLS and interpret coefficients with the controls held fixed.

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

WAGE1

Estimated time

25 to 40 min

Packages

pandas, numpy, statsmodels, patsy

Expected output

A regression or inference table with coefficients, uncertainty, and short interpretation notes.

Learning goals

  • Load and inspect WAGE1.
  • Run the Python cells connected to Multiple Regression in Python.
  • Interpret the output using multiple regression and WAGE1.

Common errors

  • File not found: check that WAGE1.DTA 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_stata("/data/WAGE1.DTA")
df.head()

Interactive activity

PrecisionSimulator

Simulate coefficient precision

Change sample size and collinearity to see what happens to standard errors.

SLEEP75

Controlled prediction

Simple prediction2.12
Multiple-regression prediction1.76
Absolute residual signal0.25

Interpretation sentence

Holding experience and tenure fixed, one more year of education changes predicted log wage by about 0.075 in this teaching setup.

Residual sketch

Variable role check

In a wage model with education as the focal x, classify experience.

Precision check

Larger samples tighten estimates. Under the Gauss-Markov assumptions, OLS has the smallest variance among linear unbiased estimators.

VIF calculator

VIF: 2.22

SE index: 0.136

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 the safest interpretation focus in Variance, Standard Errors, and Degrees of Freedom?

Quick quiz

Which interpretation habit is most important in Variance, Standard Errors, and Degrees of Freedom?

Quick quiz

Why is SLEEP75 a reasonable practice dataset here?

Key takeaway

Variance, Standard Errors, and Degrees of Freedom helps turn multiple regression output into a careful ceteris paribus interpretation.

Download dataDownload resultsDownload interpretation