IER Chapters 3.7 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/3.7-scatterplots-sample-covariance-and-sample-correlation.html

Definitions:

1. Scatterplot: A graphical representation that shows the relationship between two variables.
2. Sample Covariance: Measures the degree to which two variables change together. It is calculated as the average of the product of the deviations of each pair of data points from their respective means.
3. Sample Correlation: A standardized measure of the relationship between two variables, ranging from -1 to 1.

Formulas:

1. Sample Covariance: \[ s_{xy} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) \]

2. Sample Correlation: \[ r_{xy} = \frac{s_{xy}}{\sqrt{s_{xx} \times s_{yy}}} \] where \(s_{xx}\) and \(s_{yy}\) are the sample variances of \(x\) and \(y\) respectively.

Key Points:

• Scatterplots are useful for visually assessing the relationship between two variables.
• Sample covariance gives an idea of the direction of the relationship between two variables.
• Sample correlation standardizes the covariance and provides a measure that is independent of the units of the variables.

R Examples:

Scatterplot

# Create a scatterplot in R
x <- c(1, 2, 3, 4, 5)
y <- c(2, 4, 1, 3, 5)
plot(x, y, main="Scatterplot Example", xlab="X values", ylab="Y values")

Sample Covariance

# Calculate sample covariance in R
(sample_covariance <- cov(x, y))

[1] 1.25

Sample Correlation

# Calculate sample correlation in R
(sample_correlation <- cor(x, y))

[1] 0.5