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/4.2-estimating-the-coefficients-of-the-linear-regression-model.html#the-ordinary-least-squares-estimator

Key Concepts and Formulas

1. OLS Estimators: Ordinary Least Squares (OLS) is used to find the best-fitting line by minimizing the sum of squared errors. The OLS estimators for the slope and intercept in a simple linear regression model are:

\[ \hat{\beta}_1 = \frac{\sum_{i=1}^{n} (X_i - \overline{X})(Y_i - \overline{Y})}{\sum_{i=1}^{n} (X_i - \overline{X})^2} \] \[ \hat{\beta}_0 = \overline{Y} - \hat{\beta}_1 \overline{X} \]

2. Predicted Values and Residuals: The predicted values (\(\hat{Y}_i\)) and residuals (\(\hat{u}_i\)) are calculated as:

\[ \hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_i \] \[ \hat{u}_i = Y_i - \hat{Y}_i \]

R Examples

Here are some R examples to demonstrate the concepts:

1. Installing and Loading the AER Package

# Install the AER package (once)
#install.packages("AER")
# Load the AER package
library(AER)

2. Loading the California School Dataset

# Load the dataset
data(CASchools)

# compute STR and append it to CASchools
CASchools$STR <- CASchools$students/CASchools$teachers 

# compute TestScore and append it to CASchools
CASchools$score <- (CASchools$read + CASchools$math)/2 

3. Computing OLS Estimators Manually

# Compute beta_1_hat
beta_1_hat <- sum((CASchools$STR - mean(CASchools$STR)) * (CASchools$score - mean(CASchools$score))) / sum((CASchools$STR - mean(CASchools$STR))^2)
# Compute beta_0_hat
beta_0_hat <- mean(CASchools$score) - beta_1_hat * mean(CASchools$STR)

4. Using lm() Function for Regression Analysis

# Estimate the model
linear_model <- lm(score ~ STR, data = CASchools)

5. Seeing Regression Output

summary(linear_model)

Call:
lm(formula = score ~ STR, data = CASchools)

Residuals:
    Min      1Q  Median      3Q     Max 
-47.727 -14.251   0.483  12.822  48.540 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 698.9329     9.4675  73.825  < 2e-16 ***
STR          -2.2798     0.4798  -4.751 2.78e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 18.58 on 418 degrees of freedom
Multiple R-squared:  0.05124,   Adjusted R-squared:  0.04897 
F-statistic: 22.58 on 1 and 418 DF,  p-value: 2.783e-06

6. Scatter Plot of Data

# plot the data
plot(score ~ STR, 
     data = CASchools,
     main = "Scatterplot of Test Score and STR", 
     xlab = "STR (X)",
     ylab = "Test Score (Y)",
     xlim = c(10, 30),
     ylim = c(600, 720))

# add the regression line
abline(linear_model)