IER Chapters 4.1 Summary

In this handout, I use ChatGPT 4.0 and the KeyMate.AI Search plugin to synthesize the information from the textbook, give examples, and create R code. All prompts are exact, but some of the output may be modified for correctness.

Hanck, Arnold, Gerber, and Schmelzer (2023). Introduction to Econometrics with R. https://www.econometrics-with-r.org/index.html. This book is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Prompt

Synthesize the information in this website. Include all relevant formulas and definitions. Also create examples utilizing R to demonstrate the concepts.

https://www.econometrics-with-r.org/4.1-simple-linear-regression.html

Definitions:

1. Simple Linear Regression: A statistical method to model the relationship between a dependent variable and an independent variable using a linear equation.
2. Dependent Variable (Regressand): The variable we are trying to predict or explain.
3. Independent Variable (Regressor): The variable used to predict the dependent variable.
4. Population Regression Line: Represents the systematic relationship between the dependent and independent variables in the population.
5. Intercept: The value of the dependent variable when the independent variable is zero.
6. Slope: Represents the change in the dependent variable for a one-unit change in the independent variable.
7. Error Term: Captures the random effects and deviations between the regression line and the actual observed data.

Formula:

The linear regression model is represented as: \[ Y_i = \beta_0 + \beta_1 X_i + u_i \] Where: - \(Y_i\) is the dependent variable for observation \(i\). - \(X_i\) is the independent variable for observation \(i\). - \(\beta_0\) is the intercept of the population regression line. - \(\beta_1\) is the slope of the population regression line. - \(u_i\) is the error term for observation \(i\).

R Examples:

Creating Sample Data:

# Create sample data
STR <- c(15, 17, 19, 20, 22, 23.5, 25)
TestScore <- c(680, 640, 670, 660, 630, 660, 635)

Visualizing Data with Scatterplot:

# Create a scatterplot of the data
plot(TestScore ~ STR, ylab="Test Score")
# Add the systematic relationship to the plot
abline(a = 713, b = -3)