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

Question

BlackTom AI · Accepted Answer

Imperfect multicollinearity means regressors are highly (but not perfectly) correlated, so they are close to being linear combinations of one another but no exact linear relation exists.

1) Relevant knowledge points and concepts
- Multicollinearity concerns correlation among regressors (columns of X) in a linear regression.
- Perfect multicollinearity: an exact linear relationship among regressors → X'X is singular (noninvertible) and OLS cannot be computed.
- Imperfect (or near) multicollinearity: regressors are highly correlated but no exact linear dependence → X'X is still invertible but ill‑conditioned.
- Consequences: OLS remains unbiased (if classical assumptions hold) but estimates have large variances and are imprecise; measures like VIF and condition number detect this.

2) Step-by-step reasoning
- OLS estimator: β̂ = (X'X)^{-1} X'y. This requires X'X to be invertible.
- If regressors are only highly correlated (imperfect multicollinearity), X'X is close to singular but still invertible, so β̂ can be computed.
- High correlation inflates variances of β̂ (diagonal elements of Var(β̂) = σ^2 (X'X)^{-1}), making inference unreliable, but unbiasedness is unaffected if regressors are exogenous.

3) Intermediate/formal steps
- Perfect case: if a_j = Σ_k λ_k a_k exactly, then det(X'X) = 0 → (X'X)^{-1} does not exist → OLS impossible.
- Imperfect case: no exact equality, so det(X'X) ≠ 0 → (X'X)^{-1} exists (though large condition number).

4) Why the correct answers are correct
- a: Correct — describes high (but not exact) correlation among regressors.
- d: Correct — because X'X remains invertible under imperfect multicollinearity, so OLS estimator can still be computed.

5) Why other options are incorrect
- b: Incorrect — multicollinearity does not introduce bias into OLS (given exogeneity); it affects variance and precision, not unbiasedness, regardless of sample size.
- c: Incorrect — this describes correlated error terms across models or observations (serial correlation or correlation across equations), not multicollinearity among regressors.

6) Conclusion
Therefore the correct choices are a and d, which matches the provided answer.

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

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

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