In the Case When the Errors Are Homoskedastic and Normally $\hat { \boldsymbol { \beta } }$

question 17

Short Answer

In the case when the errors are homoskedastic and normally distributed, conditional on X, then a. $\hat { \boldsymbol { \beta } }$ is distributed $N \left( \boldsymbol { \beta } , \Sigma _ { \hat { \beta } \mid X } \right)$ , where $\Sigma _ { \hat { \beta } \mid X } = \sigma _ { u } ^ { 2 } \boldsymbol { I } _ { ( \mathrm { k } + 1 ) }$ .
b. $\hat { \boldsymbol { \beta } }$ is distributed $\mathrm { N } \left( \boldsymbol { \beta } , \Sigma _ { \hat { \beta } } \right)$ , where $\Sigma _ { \hat { \beta } } = \Sigma _ { \sqrt { n } ( \dot { \beta } - \beta ) } / n = \boldsymbol { Q } _ { X } ^ { - 1 } \Sigma _ { V } \boldsymbol { Q } _ { X } ^ { - 1 } / n$ .
c. $\hat { \beta }$ is distributed $N \left( \boldsymbol { \beta } , \Sigma _ { \dot { \beta } \mid X } \right)$ , where $\Sigma _ { \dot { \beta } \mid X } = \sigma _ { u } ^ { 2 } ( \boldsymbol { X } \boldsymbol { X } ) ^ { - 1 }$ .
d. $\hat { U } = \boldsymbol { P } _ { \boldsymbol { X } } \boldsymbol { Y }$ where $\boldsymbol { P } _ { \boldsymbol { X } } = \boldsymbol { X } \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime }$ .

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Definitions:

Continuous Problem Solving

A relentless, ongoing process of identifying, assessing, and addressing issues and problems as they arise in real-time.

Throughput

The rate at which a system generates goods or services over a certain period of time.

Reduced Inventory

Reduced inventory involves minimizing the amount of goods or materials held in stock to lower costs, reduce waste, and increase efficiency in production and distribution processes.

JIT

Just-In-Time, an inventory management strategy to increase efficiency and decrease waste by receiving goods only as they are needed in the production process.