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\begin{aligned} l (\hat{ψ}) & = - \frac{n}{2} \log ({\hat{σ}}^{2}) - \frac{1}{2} \log | V^{*} | - \frac{1}{2} (y - X \hat{β})^{'} {V^{*}}^{- 1} (y - X \hat{β}) / {\hat{σ}}^{2} - \frac{n}{2} \log (2 π) \\ l ({\hat{ψ}}_{(U)}) & = - \frac{n}{2} \log ({\hat{σ}}_{(U)}^{2}) - \frac{1}{2} \log | V^{*} | - \frac{1}{2} (y - X {\hat{β}}_{(U)})^{'} {V^{*}}^{- 1} (y - X {\hat{β}}_{(U)}) / {\hat{σ}}_{(U)}^{2} - \frac{n}{2} \log (2 π) \end{aligned}

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\begin{align*} l(\widehat{\bpsi }) & = -\frac{n}{2}\log (\widehat{\sigma }^2) - \frac12\log |\mb{V}^*| - \frac12 (\mb{y}-\mb{X}\widehat{\bbeta })'{\mb{V}^*}^{-1} (\mb{y}-\mb{X}\widehat{\bbeta }) / \widehat{\sigma }^2 - \frac{n}{2}\log (2\pi ) \\ l(\widehat{\bpsi }_{(U)}) & = -\frac{n}{2}\log (\widehat{\sigma }^2_{(U)}) -\frac12 \log |\mb{V}^*| - \frac12 (\mb{y}-\mb{X}\widehat{\bbeta }_{(U)})'{\mb{V}^*}^{-1} (\mb{y}-\mb{X}\widehat{\bbeta }_{(U)}) / \widehat{\sigma }^2_{(U)} - \frac{n}{2}\log (2\pi ) \end{align*}