Goodness of Fit: R-Squared and Adjusted R-Squared

If an additional regressor is added to a multiple regression model, the R-squared statistic will rise, and the adjusted R-squared statistic:

Incorrect. The addition of a good, appropriate regressor can greatly increase the explanatory power of the model, causing the adjusted R-squared statistic to rise.

Incorrect. Consider that the adjusted R-squared statistic includes a _k_ term in the denominator to lower its value with each new regressor, holding all else constant.

Correct! If the additional regressor is adding value, then the adjusted R-squared statistic could very well rise as additional variation is being explained. But if the new regressor adds little, the addition to the explained variation might be smaller than the adjustment made by the adjusted R-squared statistic for the extra regressor, causing that value to fall.

must fall.

must also rise.

can either rise or fall.