Three contrasts yield the same Hausman test. (a) Verify that (m_{2}) is numerically exactly identical to (m_{1})

Question:

Three contrasts yield the same Hausman test.

(a) Verify that $m_{2}$ is numerically exactly identical to $m_{1}$ and $m_{3}$, where $m_{i}=\widehat{q}_{i}^{\prime} V_{i}^{-1} \widehat{q}_{i}$ defined below (4.48).

(b) Verify that these are also exactly numerically identical to $m_{4}=\widehat{q}_{4}^{\prime} V_{4}^{-1} \widehat{q}_{4}$ where $\widehat{q}_{4}=\widehat{\beta}_{G L S}-\widehat{\beta}_{O L S}$ and $V_{4}=\operatorname{var}\left(\widehat{q}_{4}ight)$.

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