Financial Risk Manager Part 1
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Which of the following statements is INCORRECT regarding the use of R² and ar{R}^2 in multiple regression analysis?
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NNikitesh
A
An increase in the R² or ar{R}^2 always means that an added variable is statistically significant
B
A high R² or ar{R}^2 does not mean that the regressors are the true cause of the dependent variable
C
A high R² or ar{R}^2 does not necessarily indicate that you have the most relevant set of regressors, nor does a low R² or ar{R}^2 necessarily indicate the presence of inappropriate regressors
D
A high R² or ar{R}^2 does not mean that we do not have omitted variable bias
Explanation:
Explanation
Statement A is incorrect because:
-
R² always increases when a regressor is added - This is a mathematical property of R². Adding any variable (even a random one) will increase or at least not decrease R².
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ar{R}^2 (adjusted R²) does not always increase - Unlike R², adjusted R² penalizes for adding variables that don't improve the model. It only increases if the added variable improves the model more than would be expected by chance.
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Statistical significance requires a t-test - Even when R² or ar{R}^2 increases, this doesn't guarantee the added variable is statistically significant. To determine significance, we need to perform a t-test on the coefficient.
Why the other statements are correct:
- Statement B: High R² doesn't imply causality. Correlation ≠ causation.
- Statement C: High R² doesn't mean we have the best set of predictors; low R² doesn't mean we have bad predictors.
- Statement D: High R² doesn't rule out omitted variable bias. We could have high R² but still miss important variables.
Key takeaway: R² measures goodness of fit, not statistical significance or model adequacy. Always use hypothesis tests (t-tests, F-tests) to evaluate variable significance.
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