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:

Proportional Differences

Refers to the comparison of ratios or fractions, highlighting the relative difference in size or magnitude between two or more quantities.

E. H. Weber

A pioneering psychologist and physician known for his work in the somatosensory system and Weber's Law, which relates to the perceived change in a stimulus to its initial intensity.

Absolute Differences

The minimum amount by which two sensory stimuli can differ for an individual to perceive them as different.

Gustav Fechner

A German philosopher, physicist, and experimental psychologist who founded psychophysics, studying the relationship between the physical properties of stimuli and the sensations they produce.