What is a risk of high but imperfect correlation between two predictors in an otherwise estimable regression?
Correct Answer
C) Separate coefficient estimates can become imprecise and unstable
High but imperfect correlation among predictors can inflate coefficient standard errors and make their separate estimated effects unstable. R-squared can remain high. This differs from exact linear dependence, which makes separate coefficients unidentified in the usual full-rank model. Predictor correlation does not determine which variable is the response.
Why This Is the Correct Answer
Option C is correct because when two predictors are highly but imperfectly correlated, the model struggles to separate their individual effects. This inflates the coefficient standard errors, making the separate estimates imprecise and unstable, so small data changes can shift them noticeably. The overall model can still fit well.
Why the Other Options Are Wrong
Option A: The model’s R-squared drops sharply toward zero as a result
Option A says the model's R-squared drops sharply toward zero as a result. Multicollinearity mainly affects the individual coefficients; R-squared can remain high even when the separate estimates are unstable.
Option B: The response variable can no longer be selected
Option B says the response variable can no longer be selected. Correlation between predictors does not determine or prevent the choice of the response variable.
Option D: The regression can no longer be computed from the sample data
Option D says the regression can no longer be computed from the sample data. That describes exact linear dependence; with high but imperfect correlation the regression can still be computed, though the separate coefficients are less reliable.
Twins confuse the model
Highly correlated predictors are like identical twins: the model can tell the pair matters but struggles to credit each one separately.
How to use: Use this image to recall that multicollinearity destabilizes individual coefficients, not the whole fit.
Exam Tip
Separate high correlation (coefficients unstable, model still runs) from perfect correlation (separate coefficients cannot be estimated).
Common Mistakes to Avoid
- -Assuming multicollinearity lowers R-squared sharply
- -Confusing high correlation with exact linear dependence
- -Trusting individual coefficients without checking their standard errors
Concept Deep Dive
Analysis
This question tests understanding of multicollinearity that is high but not perfect. The regression can still be estimated, but correlated predictors share explanatory information, which makes it hard to separate their individual effects. Candidates must distinguish this from exact linear dependence and from effects on overall fit.
Background Knowledge
Multicollinearity occurs when predictors in a regression are strongly related to each other. When the relationship is exact, separate coefficients cannot be identified; when it is high but imperfect, they can be estimated but with large standard errors. Appraisers see this, for example, when living area and bedroom count move closely together.
Real-World Application
An appraiser's regression includes both gross living area and number of rooms, which move closely together. The room coefficient flips sign when a few sales are removed, so she drops one variable and re-checks the model.
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