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Interpret the goodness-of-fit measures R2 and adjusted-R2 for both single and multiple regressions.

FRM® Part I · Quantitative Analysis · Regression with Multiple Explanatory Variables · FRM1.QA.8.c

The idea

R² is the fraction of the variation in Y that is explained by the regression. R² = 1 − SSR ÷ TSS. Here SSR is the sum of squared residuals and TSS is the total sum of squares of Y. In a simple regression it equals the squared correlation between X and Y. In any OLS regression with an intercept it equals the squared correlation between Y and its fitted values. Adding a regressor never lowers R², so R² cannot be used to choose among models with different numbers of variables. Adjusted R² is computed from R², the number of observations n and the number of regressors k: Adjusted = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1). It penalises extra regressors, is below R² when k is at least 1 and R² is below 1, and can be negative. It rises when an added variable has an absolute t-statistic above one. Neither measure shows that coefficients are unbiased or that the model is correct. A common mistake is using n − k instead of n − k − 1 as the denominator.

Formulas

Adjusted R²

Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1)

R²
the unadjusted coefficient of determination
n
the number of observations
k
the number of regressors, not counting the intercept

Comparing models with different numbers of regressors on the same data. The trap is using n − k in the denominator instead of n − k − 1.

A practice question

From the question bank, with the reason each wrong option is wrong.

A variable is added to an OLS regression estimated on the same sample. Which statement about R2 and adjusted R2 is correct?

  1. R2 cannot fall, but adjusted R2 can fall if the variable adds too little - correct
  2. R2 can fall if the variable is irrelevant, but adjusted R2 never fallsWhy not: Reverses the roles: R2 never falls when a regressor is added, while adjusted R2 includes a penalty and can fall.
  3. Both R2 and adjusted R2 fall if the variable is irrelevantWhy not: OLS cannot make the sum of squared residuals larger by adding a regressor, so R2 does not fall.
  4. R2 rises only if the new coefficient is significant at the 5 percent levelWhy not: R2 rises (weakly) whatever the significance of the new coefficient.

Adding a regressor can only lower or keep the residual sum of squares, so R2 never decreases. Adjusted R2 = 1 - (1 - R2)(n - 1)/(n - k - 1) penalises the loss of degrees of freedom and falls when the added variable contributes little.

A harder one

A regression with 5 explanatory variables and 30 observations has R2 = 0.62. The adjusted R2 is closest to:

  1. 0.559Why not: Uses n - k instead of n - k - 1 as the residual degrees of freedom.
  2. 0.541 - correct
  3. 0.251Why not: Applies the degrees-of-freedom ratio to R2 instead of to 1 - R2.
  4. 0.620Why not: Reports the unadjusted R2; the adjustment lowers it when k is greater than zero.

Adjusted R2 = 1 - (1 - 0.62) x (30 - 1)/(30 - 5 - 1) = 1 - 0.38 x 29/24 = 0.541.

The hardest tier

A model with 3 regressors fitted to 40 observations has R2 = 0.700. Two more regressors are added and R2 becomes 0.706. After the change, the adjusted R2 is closest to:

  1. 0.706, higher than beforeWhy not: Reads the unadjusted R2; the adjusted figure must account for the two lost degrees of freedom.
  2. 0.672, lower than beforeWhy not: Uses n - k instead of n - k - 1 in the denominator.
  3. 0.663, lower than before - correct
  4. 0.690, higher than beforeWhy not: Counts only the two added regressors as k, rather than all five.

Before: 1 - 0.300 x 39/36 = 0.675. After: 1 - (1 - 0.706) x 39/(40 - 5 - 1) = 1 - 0.294 x 39/34 = 0.663. The small gain in R2 does not offset the penalty, so adjusted R2 falls.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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