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FRM Part I · FRM Exam Part I · Regression with Multiple Explanatory Variables

An analyst adds a fifth explanatory variable to a regression estimated on 40 observations. Which statement about the effect on R-squared and adjusted R-squared is correct?

R-squared never decreases when a regressor is added, because the residual sum of squares cannot rise. Adjusted R-squared includes a degrees-of-freedom penalty and can fall when the new variable contributes little explanatory power, so it can decrease.

  1. AR-squared cannot decrease, while adjusted R-squared can decreaseCorrect
  2. BR-squared can decrease, while adjusted R-squared cannot decrease
  3. CBoth R-squared and adjusted R-squared cannot decrease
  4. DBoth R-squared and adjusted R-squared must increase if the variable's coefficient is nonzero

Explanation

Adding a regressor to an OLS model with an intercept never raises RSS, so R-squared cannot fall. Adjusted R-squared penalizes the lost degree of freedom, so it falls if the new variable adds too little explanatory power (absolute t-statistic below 1). Therefore only adjusted R-squared can decrease.

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