Prove That Under the Extended Least Squares Assumptions the OLS $\hat { \beta}$

question 38

Essay

Prove that under the extended least squares assumptions the OLS estimator $\hat { \beta}$ is unbiased and that its variance-covariance matrix is $\sigma _ { u } ^ { 2 }$ (X'X)^-1.

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Definitions:

Sustained Firsthand Contact

Direct and ongoing engagement with people, environments, or experiences.

Integration

The process of combining separate parts or elements to form a coherent whole, often used in contexts of social, system, or data consolidation.

Anterior Prefrontal Cortex

A region of the brain located at the front of the frontal lobes, associated with complex behaviors and decision making.

Functional Magnetic Resonance Imaging

A neuroimaging technique that measures brain activity by detecting changes associated with blood flow, used to observe brain structures and functions.