An intercept OLS regression uses 12 sales and 8 predictors, with a full-rank design. What is the principal limitation?
Correct Answer
B) Only 3 residual degrees of freedom remain, limiting error assessment
Eight predictors plus an intercept require nine coefficients. With twelve observations and a full-rank design, residual degrees of freedom=12−9=3. This leaves limited information for estimating error and assessing generalization, raising instability and overfitting concerns. It does not force all coefficients to zero or make R-squared undefined. No single observations-per-variable rule guarantees reliability.
Why This Is the Correct Answer
Option B is correct because eight predictors plus an intercept require nine coefficients, leaving 12 − 9 = 3 residual degrees of freedom in a full-rank design. So few degrees of freedom limit the ability to estimate error and assess how well the model generalizes.
Why the Other Options Are Wrong
Option A: The response variable is necessarily misidentified in the model
Option A says the response variable is necessarily misidentified, but the number of predictors relative to sales has nothing to do with which variable is the response.
Option C: R-squared cannot be computed at all for this particular set of sales data
Option C says R-squared cannot be computed at all, but R-squared can still be calculated with 3 residual degrees of freedom. The concern is that it may look strong because of overfitting.
Option D: Every fitted coefficient in the model must equal exactly zero
Option D says every coefficient must equal exactly zero, but nothing about the sample size forces coefficients to zero. The coefficients are estimated, though they may be unstable.
Sales minus coefficients
Residual degrees of freedom = sales minus coefficients, counting the intercept. Twelve minus nine leaves three.
How to use: Use this to check whether a regression has enough data for its number of predictors.
Exam Tip
Remember to count the intercept as a coefficient when computing residual degrees of freedom.
Common Mistakes to Avoid
- -Forgetting to count the intercept
- -Thinking R-squared cannot be computed with few degrees of freedom
- -Trusting a high R-squared from an overfitted model
Concept Deep Dive
Analysis
This question tests degrees of freedom in regression. With an intercept and eight predictors, the model fits nine coefficients; with twelve sales, only three residual degrees of freedom remain. That leaves little information for estimating error and raises concerns about instability and overfitting.
Background Knowledge
Residual degrees of freedom equal the number of observations minus the number of fitted coefficients, including the intercept. Few residual degrees of freedom make error estimates and significance tests weak and increase the risk of overfitting. No single observations-per-variable rule guarantees reliability, but more data relative to predictors generally helps.
Real-World Application
An appraiser tries to fit a regression with eight variables to twelve local sales. Seeing only three residual degrees of freedom, she reduces the number of variables and expands the data set before relying on the results.
More Statistics Questions
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