Hausman (1978) test based on an artificial regression. Show that Hausman's test can be alternatively obtained from any one of the following artificial regressions:

\[\begin{aligned}& y^{*}=X^{*} \beta+\tilde{X} \gamma+w_{1} \\& y^{*}=X^{*} \beta+\bar{X} \gamma+w_{2} \\& y^{*}=X^{*} \beta+X \gamma+w_{3}\end{aligned}\]

where \(y^{*}=\sigma_{u} \Omega^{-1 / 2} y, X^{*}=\sigma_{u} \Omega^{-1 / 2} X, \widetilde{X}=Q X\), and \(\bar{X}=P X\); see (4.42). Hausman's test is equivalent to testing whether \(\gamma=0\) from any one of these three OLS regressions; see Baltagi and Liu (2007).

y* = X*B+Xy+w W (4.42)