2 marks, −0.667EconometricsGoodness of fit and model selectionAdjusted R-squared and model comparisonModerateabout 1 min

IIT JAM 2022 Question 27: Goodness of fit and model selection

Which of the following is NOT correct regarding R-squared $(R^2)$ and adjusted R-squared $(\bar R^2)$ ?

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Answer and worked solution

Answer: (B)

$R^2$ is scale invariant (A), it cannot fall when a regressor is added (C), and the formula in (D) is the standard definition of $\bar R^2$ .

Adjusted $R^2$ penalises extra parameters and can turn negative when regressors add little explanatory power, so the claim that it is always positive is false. The answer is (B).

The trap: (C)

R^2

tends to increase if we add an additional explanatory variable.

Statement (C) sounds like a flaw but is a true property of $R^2$ ; the false statement is the guaranteed positivity of $\bar R^2$ .

Plain-text version

Which of the following is NOT correct regarding R-squared (R²) and adjusted R-squared (R̄²)? (A) R² is a scale invariant statistic. (B) R̄² is always positive. (C) R² tends to increase if we add an additional explanatory variable. (D) R̄² = 1 - (1 - R²)((n-1)/(n-k)), where k is the number of parameters and n the number of observations.

Also written as: Which of the following is NOT correct regarding R-squared (R^2) and adjusted R-squared (R_bar^2)?

Goodness of fit and model selection has come up 27 times in 6 exams since 2009.

What this question tests

This question is on Adjusted R-squared and model comparison, part of Goodness of fit and model selection in Econometrics.

Penalising extra regressors and when R-squared can be compared across models.

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