Lesson 7

Fitted Values and Residuals

Big question

How do fitted values and residuals work with several x variables?

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain fitted values and residuals in plain language.
  • Use ceteris paribus correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

A fitted value uses every estimated coefficient and every explanatory variable in the row. The residual is actual y minus fitted y.

Key terms

ceteris paribus
Other included factors held fixed for interpretation.
omitted-variable bias
Bias that can arise when a relevant omitted factor is related to an included regressor.
multicollinearity
Strong but not perfect relationships among explanatory variables.

Core formula

uhati=yi−yhatiuhat_i = y_i - yhat_i

Use plain-language interpretation before algebra.

Example

A positive residual means the model underpredicted the observation. A negative residual means it overpredicted.

Interactive visual

Residual calculator and observation table

Original Module 3 visual for Fitted Values and Residuals.

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

Fitted Values and Residuals Python example

Fitted Values and Residuals 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, matplotlib

Expected output

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

Learning goals

  • Load and inspect WAGE1.
  • Run the Python cells connected to Fitted Values and Residuals.
  • Interpret the output using fitted values and residuals.

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

PartiallingOutVisualizer

Inspect fitted values and residuals

Read actual, fitted, and residual values for several observations.

HPRICE1

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.

Control overlap

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 Fitted Values and Residuals?

Quick quiz

Which interpretation habit is most important in Fitted Values and Residuals?

Quick quiz

Why is HPRICE1 a reasonable practice dataset here?

Key takeaway

Fitted Values and Residuals helps turn multiple regression output into a careful ceteris paribus interpretation.

Download dataDownload resultsDownload interpretation