Define the GLS Estimator and Discuss Its Properties When Ω $x _ { 1 i } ^ { 2 }$

question 48

Essay

Define the GLS estimator and discuss its properties when Ω is known. Why is this estimator sometimes called infeasible GLS? What happens when Ω is unknown? What would the Ω matrix look like for the case of independent sampling with heteroskedastic errors, where var(u_i | X_i)= ch(X_i)= σ²
$x _ { 1 i } ^ { 2 }$ ? Since the inverse of the error variance-covariance matrix is needed to compute the GLS estimator, find Ω^-1. The textbook shows that the original model Y = Xβ + U will be transformed into $\widetilde { Y } = \widetilde { X } \beta + \widetilde { U } , \text { where } \widetilde { Y } = F Y , \widetilde { X } = F X \text {, and } \widetilde { U }$ = FU, and $F$ F = Ω^-1. Find F in the above case, and describe what effect the transformation has on the original data.

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Definitions:

Fixed Costs

Financial obligations like rent, wages, and insurance that are stable and do not vary with the level of goods manufactured or sold.

Break-even

The point at which total revenues equal total costs, resulting in neither profit nor loss for the business.

Margin of Safety

The difference between actual or projected sales and the break-even point. It indicates the amount of sales decline a business can endure before it starts incurring losses.

Variable Costs

Costs that change in proportion to the level of activity or production volume.