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Answer: Model 2 must have a lower value of adjusted R-squared
When two regression models have identical R-squared values, it means the additional variable $ x_3 $ has zero explanatory power for $ y $ and its coefficient estimate $ \beta_3 $ must be exactly zero. The adjusted R-squared formula is: $$\bar{R}^2 = 1 - \left( \frac{n - 1}{n - k - 1} \right)(1 - R^2)$$ where $ k $ represents the number of regressors. Since Model 2 has more regressors (k increases from 2 to 3) while R-squared remains the same, the denominator $(n - k - 1)$ decreases, making the fraction $\frac{n - 1}{n - k - 1}$ larger. This causes the adjusted R-squared for Model 2 to be lower than for Model 1. Therefore, Model 2 must have a lower value of adjusted R-squared.
Author: Nikitesh Somanthe
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Consider the following 2 regression models:
Model 1:
Model 2:
A researcher determines that the two models have identical R-squared values. This most likely implies that:
A
Model 2 must have a lower value of adjusted R-squared
B
Model 2 must have a higher value of adjusted R-squared
C
Model 1 and 2 will also have identical values of adjusted R-squared
D
Variable is statistically significant