1.6.1.md

Solution to Review Question

by Qiang Gao, updated at May 15, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 6 Generalized Least Squares (GLS)

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Review Question 1.6.1 (The no-multicollinearity assumption for the transformed model)

Assumption 1.3 for the transformed model is that $$ \mathrm{rank} ( \mathbf{C} \mathbf{X} ) = K $$. This is satisfied since $$ \mathbf{C} $$ is nonsingular and $$ \mathbf{X} $$ is of full column rank. Show this.

Solution

To show $$ \mathrm{rank} ( \mathbf{C} \mathbf{X} ) = K $$ is equivalent to show $$ \mathbf{C} \mathbf{X} \mathbf{v} \neq \mathbf{0} $$ for any $$ \mathbf{v} \neq \mathbf{0} $$.

For any $$ \mathbf{v} \neq \mathbf{0} $$, $$ \mathbf{X} \mathbf{v} \neq \mathbf{0} $$ because $$ \mathbf{X} $$ is of full column rank. Let $$ \mathbf{u} \equiv \mathbf{X} \mathbf{v} \neq \mathbf{0} $$, because $$ \mathbf{C} $$ is nonsingular (also full column rank), $$ \mathbf{C} \mathbf{u} = \mathbf{C} \mathbf{X} \mathbf{v} \neq \mathbf{0} $$. Q.E.D.

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Copyright ©2017 by Qiang Gao