Add a worthless variable to a regression and R² still goes up. That single fact is why the exam tests adjusted R², information criteria, and the joint F-test instead.
The coefficient of determination (R²) is the ratio of the sum of squares regression (SSR) to the sum of squares total (SST). In multiple regression it is a weak fit measure: adding any independent variable makes R² rise or stay flat, never fall. R² tells you nothing about whether coefficients are significant, whether estimates are biased, or whether the model is good. A sound asset-pricing model can have low R²; an overfit model, one carrying too many variables relative to observations, can have high R² and useless coefficients.
Adjusted R² penalizes for degrees of freedom, where k is the number of independent variables and n the number of observations.
KEY: Adjusted R² rises when the added variable's |t-statistic| exceeds 1.0 and falls when it is below 1.0.
Common mistakes
- Judging models by R². It never falls when variables are added, so it always flatters the biggest model. Compare adjusted R², AIC, and BIC instead.
- Reading adjusted R² as a percentage explained. That interpretation dies in multiple regression, and the statistic can even go negative.
- Comparing AIC across different dependent variables. The criteria are valid only within a set of models explaining the same Y, and only relatively.
Bottom line
- R² never decreases as variables are added; it cannot show significance, bias, or true fit quality
- Adjusted R² = 1 − (n−1)/(n−k−1); it rises only if the added variable's |t| > 1.0
- Adjusted R² is below R² whenever k ≥ 1 and may be negative
- AIC and BIC: lower is better, relative values only, same dependent variable required
Exam shortcut
Read the ANOVA table first and pull three numbers: SSR, SSE, SST. R² is SSR/SST; MSR is SSR/k; MSE is SSE/(n − k − 1). Every other statistic falls out of those. Stem signals map cleanly. "Does the model explain anything" means MSR/MSE with k and n − k − 1. "Do these two variables jointly matter" means the restricted-minus-unrestricted F with q in the numerator.
The full lesson (about 2,021 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- regression model fit
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