Choose ALL CORRECT statement(s) about the imperfect multicollinearity:

Question

BlackTom AI · Accepted Answer

Imperfect multicollinearity means two or more regressors are highly (but not perfectly) correlated, so the design matrix X has columns that are nearly linearly dependent.

1) Relevant concepts
- Multicollinearity refers to correlation among regressors. Perfect multicollinearity: an exact linear relationship among regressors => X'X is singular and (X'X)^{-1} does not exist. Imperfect (or near) multicollinearity: regressors are highly correlated but no exact linear dependence => X'X is invertible but ill-conditioned. Under the classical linear model assumptions, OLS remains unbiased; multicollinearity inflates variances of coefficient estimates and makes them imprecise.

2) Step-by-step reasoning
- OLS estimator: β̂ = (X'X)^{-1}X'y. 
- If multicollinearity is imperfect, X'X is not singular, so the inverse exists and β̂ can be computed.
- However, because X'X is nearly singular (ill-conditioned), the elements of (X'X)^{-1} become large, which increases Var(β̂) = σ^2 (X'X)^{-1}. Large variances lead to wide confidence intervals and unstable coefficient signs/magnitudes.

3) Calculation / intermediate step
- Show dependence on invertibility: compute β̂ requires (X'X)^{-1}. Imperfect multicollinearity: det(X'X) ≠ 0 (so invertible). Perfect multicollinearity: det(X'X) = 0 (no inverse).

4) Why the correct answers are correct
- Option b is correct because it states the definition: high correlation among regressors.
- Option c is correct because with imperfect (not perfect) multicollinearity the OLS estimator can still be computed since X'X is invertible.

5) Why other options are incorrect
- Option a is incorrect: multicollinearity does not make OLS biased; OLS remains unbiased under standard assumptions regardless of sample size—only variance is affected. A larger sample size does not change the unbiasedness property.
- Option d is incorrect: it describes correlated error terms across models (cross-equation error correlation), not multicollinearity among regressors.

6) Conclusion
Both b and c are correct, matching the provided answer.

Choose ALL CORRECT statement(s) about the imperfect multicollinearity:多项选择题

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