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Answer to part b
Consider a regression model written in vector form \[ \unde ...
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Answer to part b
Consider a regression model written in vector form \[ \underset{T \times 1}{Y}=\underset{T \times K}{X} \underset{K \times 1}{\beta}+\underset{T \times 1}{\varepsilon}, \quad \mathbb{E}[\varepsilon]=\underset{T \times 1}{0}, \quad \operatorname{Var}(\varepsilon)=\sigma^{2} \mathrm{Id}_{T}, \] where \( X \) is deterministic. a) Show that the OLS estimator \( \widehat{\beta}=\left(X^{\prime} X\right)^{-1} X^{\prime} Y \) is unbiased and derive its variance [10 marks] b) Consider a linear estimator \( \widetilde{\beta} \) of \( \beta \). Show that \( \widetilde{\beta} \) can be written as \( \widetilde{\beta}=\left(\left(X^{\prime} X\right)^{-1} X^{\prime}+D^{\prime}\right) Y \), where \( D^{\prime} \) is \( K \times T \). Which \( D \) should be used to ensure that \( \widetilde{\beta}=\widehat{\beta} \) ? Show that \( \widetilde{\beta} \) is unbiased if and only \( D^{\prime} X=0 \).