4. Putting It All Together: A Glimpse of Simple Linear Regression
Simple linear regression is a fundamental econometric tool that models the relationship between a dependent variable (like a stock’s return) and an independent variable (like the market’s return). It uses the concepts we’ve discussed—mean, variance, and correlation—to build a predictive model.
4.1. The Characteristic Line: Modeling a Stock’s Return
One of the most common applications of simple linear regression in finance is estimating a stock’s characteristic line. This model describes how a stock’s return is related to the return of the overall market. For General Electric (GE) stock and the S&P 500 market index, the model is:
Return on GE stock = α + β (Return on S&P 500) + Error term
The model’s key parameters, Alpha (α) and Beta (β), have very specific financial interpretations.
| Parameter | Financial Interpretation |
| Beta (β) | The slope of the regression line. It measures the sensitivity of a stock’s return to the overall market’s return. A beta greater than 1 means the stock is more volatile than the market. |
| Alpha (α) | The intercept of the regression line. It is interpreted as a measure of a stock’s performance above or below what would be expected based on the market’s performance and is often called the “average abnormal performance.” |
For this specific example, the source text estimates the parameters using historical data, yielding the following characteristic line: Return on GE stock = 0.0093 + 1.0575 (Return on S&P 500). This tells us two things: (1) GE’s beta of 1.0575 indicates it was slightly more volatile than the market during this period, and (2) its small positive alpha of 0.0093 suggests a minor ‘abnormal performance’ above what the market’s return would predict.
4.2. R-Squared: How Good is the Fit?
After building a regression model, we need to know how well it actually explains the data. The coefficient of determination (R²) is a measure of the “goodness-of-fit” for the regression. In simple terms, it represents the percentage of the variation in the dependent variable (GE’s return) that can be explained by the variation in the independent variable (the S&P 500’s return).
A key property of R² is that it is simply the squared correlation coefficient between the two variables. For the GE/S&P 500 model, the R² is 0.5076. This means that approximately 51% of the monthly variation in GE’s stock return can be explained by the monthly variation in the S&P 500’s return.