Multicollinearity in a regression model occurs when:
Correct Answer
B) Two or more predictors are strongly linearly related to each other
Multicollinearity is substantial linear dependence among predictors. High pairwise correlation is one example, though combinations of several predictors can be involved without extreme pairwise correlations. It can make individual coefficients imprecise or unstable. Predictor count alone does not define it, and randomly scattered residuals do not establish absence of predictor dependence.
Why This Is the Correct Answer
Option B is correct because multicollinearity is substantial linear dependence among predictors, such as two or more predictors that are strongly linearly related to each other. High pairwise correlation is one example, though a combination of several predictors can also be involved. It can make individual coefficients imprecise or unstable.
Why the Other Options Are Wrong
Option A: The response variable has too few observations for the model
Option A describes the response variable having too few observations, which is a sample-size problem, not multicollinearity. Multicollinearity concerns linear dependence among the predictors themselves.
Option C: The model includes more than five predictor variables
Option C says multicollinearity occurs when the model includes more than five predictors, but predictor count alone does not define it. A model with two predictors can be highly collinear, while a model with many unrelated predictors may not be.
Option D: Residuals are randomly scattered around zero in a residual plot
Option D describes residuals randomly scattered around zero, which relates to error assumptions, not to dependence among predictors. A good-looking residual plot does not establish that the predictors are free of linear dependence.
Co-linear Means Predictors Collide
The prefix 'multi-co-linear' points to multiple predictors lying along the same line together, so their separate effects collide.
How to use: Use the word itself as the clue: if an option is not about predictors relating to each other, it is not multicollinearity.
Exam Tip
Separate predictor problems (multicollinearity) from residual problems (heteroscedasticity, nonlinearity) and sample-size problems.
Common Mistakes to Avoid
- -Thinking multicollinearity is defined by having many predictors
- -Confusing multicollinearity with residual pattern problems
- -Assuming it involves the response variable rather than the predictors
Concept Deep Dive
Analysis
This question tests the definition of multicollinearity. The key is that it concerns relationships among the predictor (independent) variables, not the response variable, the number of predictors or the residual pattern. Recognizing that it is about predictors being linearly related to one another identifies the answer.
Background Knowledge
In multiple regression, each coefficient estimates a predictor's effect while holding others constant. When predictors move together, the model has difficulty separating their individual effects, which inflates standard errors and can make coefficients unstable or give them unexpected signs, even if overall fit looks good.
Real-World Application
An appraiser models price using both living area and number of bedrooms and finds the bedroom coefficient negative and unstable. Because larger homes tend to have more bedrooms, the two predictors are strongly related, a classic case of multicollinearity.
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