Simple linear regression models the relationship between two quantitative variables.

• Response variable: the variable we want to explain or predict
• Explanatory variable: the variable used to explain changes in the response
• A straight line is fitted to describe the relationship between the two variables
• In finance, regression can be used to study how a stock’s returns relate to market returns

The simple linear regression model is:

\[ R_i = \alpha + \beta R_m + \epsilon \]

where:

• \(R_i\) = return of the individual stock
• \(R_m\) = return of the overall market
• \(\alpha\) = intercept
• \(\beta\) = sensitivity of the stock’s return to the market return
• \(\epsilon\) = random error term

We will use simulated daily return data to demonstrate simple linear regression.

• Market Return: daily percentage return of the overall market
• Stock Return: daily percentage return of an individual stock
• We will examine whether changes in market returns help explain changes in stock returns
• A positive relationship would indicate that the stock tends to move in the same direction as the market

## `geom_smooth()` using formula = 'y ~ x'

The fitted regression equation can be written as:

\[ \hat{R}_i = \hat{\alpha} + \hat{\beta}R_m \]

• \(\hat{R}_i\) is the predicted stock return
• \(\hat{\alpha}\) is the estimated intercept
• \(\hat{\beta}\) is the estimated slope
• A positive \(\hat{\beta}\) indicates that stock and market returns tend to move in the same direction
• The distance between an observed return and its predicted return is called a residual

The following R code creates the fitted regression plot:

ggplot(returns, aes(x=Market_Return, y=Stock_Return)) +
  geom_point() +
  geom_smooth(method="lm", se=FALSE) +
  xlab("Market Return (%)") +
  ylab("Stock Return (%)") +
  theme_bw()

• Simple linear regression measures the relationship between two quantitative variables
• In our example, stock returns have a positive relationship with market returns
• The slope measures how strongly the stock responds to changes in the market
• Regression plots help visualize both the observed data and the fitted relationship
• Interactive Plotly graphs provide another way to explore the relationship