(Requires Appendix Material) Your Textbook Shows That OLS Is a Linear

question 12

Essay

(Requires Appendix material) Your textbook shows that OLS is a linear estimator
$\hat { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } \hat { a } _ { i } Y _ { i } , \text { where } \hat { a } _ { i } = \frac { X _ { i } - \bar { X } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$ .
For OLS to be conditionally unbiased, the following two conditions must hold:
$\sum _ { i = 1 } ^ { n } \hat { a } _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } \hat { a } _ { i } X _ { i } = 1$
Show that this is the case.

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Definitions:

Bonds Sold

Refers to bonds that have been issued by an entity and purchased by investors, representing a loan from the investors to the entity.

Bond Payable

A long-term liability where the borrower is obliged to pay back a specified sum of money at a future date, including interest payments.

Accrual of Interest

The process of recognizing interest earned or incurred that has not yet been paid or received, reflecting the economic impact of interest in the accounting period it relates to.

Semiannual Interest Expense

The cost incurred for borrowing money, calculated and paid every six months.