Which of the following are consequences of...

Which of the following are consequences of heteroskedasticity?
i. The OLS estimators, $\hat{\beta}_{j},$ are inconsistent.
ii. The usual $F$ statistic no longer has an $F$ distribution.
iii. The OLS estimators are no longer BLUE.

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Key Concepts

Implications for Hypothesis Testing

The presence of heteroskedasticity affects the reliability of standard hypothesis tests, such as the F test. Because the usual formulas for calculating test statistics assume homoskedasticity, when this assumption is violated, the test statistics (like the F statistic) may follow distributions that differ from their theoretical ones, leading to invalid inference unless corrected for heteroskedasticity.

Heteroskedasticity

Heteroskedasticity occurs when the variance of the error terms in a regression model is not constant across observations. This violates one of the key classical assumptions of regression analysis, potentially leading to inefficient estimates and unreliable inference if standard errors are not adjusted for the varying variance.

Ordinary Least Squares (OLS) Properties

Under heteroskedasticity, while the OLS estimators remain unbiased and consistent, they lose their efficiency. This means that although the estimators converge to the true parameter values and do not systematically err, they do not have the smallest possible variance among linear unbiased estimators, and thus are no longer considered BLUE (Best Linear Unbiased Estimators).

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