One factor rarely explains a stock's return. Add three, and the arithmetic gets easy but the interpretation gets subtle: every slope now answers a conditional question, not a standalone one.
Multiple linear regression explains the variation in one dependent variable using two or more independent variables. Analysts use it three ways: to identify relationships between variables, to test existing theories, and to forecast. Typical investment applications include estimating a Fama and French five factor model to see which factors drive a stock's excess return, predicting financial distress from leverage, profitability, revenue growth, and market share changes, and measuring how country risk dimensions such as political stability, economic conditions, and environmental, social, and governance (ESG) factors affect equity returns.
The analyst specifies the model; software estimates it. The sequence is fixed and testable:
- Specify. Choose the dependent variable and the independent variables, which may be continuous (financial ratios) or discrete dummy variables (industry membership).
- Choose the model type. If the dependent variable is continuous, such as returns, use ordinary least squares (OLS).
Common mistakes
- Dropping "holding others constant." Interpreting 0.5892 as the total effect of the market ignores that it is a partial effect measured with SMB and HML fixed.
- Judging a variable by its scatterplot. SMB can look flat against returns pairwise and still be significant in the regression; the pairs plot cannot show partial effects.
- Using the wrong plot for the wrong assumption. A pairwise scatterplot detects linear relations, not heteroskedasticity or normality. The Q-Q plot answers only normality.
Bottom line
- Uses: identify relationships, test theories, forecast; examples include factor models, distress prediction, country-risk effects on returns
- Process: specify, choose model type, estimate, check assumptions and residuals, test fit, compare models, then adjust or deploy
- Continuous dependent variable uses OLS; discrete 1/0 dependent variable uses logistic regression
- Model: Y = b0 + b1X1 +... + bkXk + error, estimated over n observations with n greater than k
Exam shortcut
Match the plot to the assumption before touching the answers. "Pairwise scatterplot" means linearity and outliers, never heteroskedasticity or normality. "Normal Q-Q" means normality only, and deviation past ±2 standard deviations means fat tails. "Residuals versus predicted" or "versus an X" means homoskedasticity, independence, and misspecification.
The full lesson (about 1,656 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- multiple regression basics
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