Adjusted R-squared differs from R-squared in that adjusted R-squared:
Correct Answer
C) Penalizes the model for variables that do not help
Why this is correct: Adjusted R-squared penalizes the model for variables that do not help. The governing concept is that R-squared always increases when a new variable is added, even if it's irrelevant, which can lead to overfitting. Adjusted R-squared corrects this by incorporating a penalty for adding variables that do not improve the model's explanatory power, providing a more accurate measure of goodness-of-fit for multiple regression. Why the other choices are wrong: 'Measures the sample's average error' is wrong because that describes a metric like the standard error of the estimate, not R-squared. 'Always exceeds the unadjusted figure' is wrong because adjusted R-squared is typically lower than or equal to R-squared; it only increases if a new variable adds sufficient explanatory power. 'Applies only to single-variable regression models' is wrong because adjusted R-squared is specifically designed for and most useful in models with multiple independent variables. Exam tip: Remember, if adding a variable doesn't help, adjusted R-squared goes down, while plain R-squared never does. This is the key distinction.
Why This Is the Correct Answer
Adjusted R-squared penalises the statistic for each additional variable relative to sample size, so it rises only where a variable improves fit more than chance would predict and can fall where a variable adds nothing.
Why the Other Options Are Wrong
Option A: Measures the sample's average error
Average error is measured by the standard error of the estimate, a separate statistic.
Option B: Always exceeds the unadjusted figure
Adjusted R-squared is always less than or equal to R-squared, never greater.
Option D: Applies only to single-variable regression models
The adjustment is most relevant in multiple regression, where the number of variables varies. It does not apply only to single-variable models.
Pay for Every Variable
Pay for Every Variable. Adjusted R-squared charges you for each one and only pays back if it helps.
How to use: Compare models on adjusted R-squared, not R-squared. The unadjusted figure always favours the bigger model.
Exam Tip
Adjusted R-squared falling when a variable is added is direct evidence that the variable does not belong.
Common Mistakes to Avoid
- -Comparing models on unadjusted R-squared
- -Believing adjusted R-squared can exceed R-squared
- -Adding variables to improve apparent fit
Concept Deep Dive
Analysis
R-squared measures the proportion of variation in the dependent variable explained by the model, and it has a structural flaw: adding any variable, however useless, can only increase it or leave it unchanged, never reduce it. Left unchecked, that property rewards throwing variables at a model until the fit looks impressive while the model becomes overfitted β tuned to the noise in one particular sample and useless for prediction. Adjusted R-squared corrects for this by penalising the statistic for each additional variable relative to the sample size, so it rises only when a new variable improves the fit by more than would be expected from chance. It can and does fall when a variable adds nothing, which makes it the appropriate measure when comparing models with different numbers of variables. For an appraiser building a regression from a limited number of sales, the distinction is practical: adjusted R-squared is what indicates whether the model has genuine explanatory power.
Background Knowledge
R-squared measures explained variation and never decreases as variables are added. Adjusted R-squared penalises for the number of predictors relative to sample size, making it appropriate for comparing models of differing complexity.
Real-World Application
An appraiser comparing a four-variable and a seven-variable model finds adjusted R-squared falls with the larger one, and adopts the simpler specification.
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