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\begin{aligned} - 2 \log L_{d} (Y, θ θ) & = N_{0} \log 2 π + \sum_{t = 1}^{n} \sum_{i = 1}^{q * p_{t}} (\log F_{t, i} + \frac{ν_{t, i}^{2}}{F_{t, i}}) - b_{n, p_{n}}^{^{'}} S_{n, p_{n}}^{- 1} b_{n, p_{n}} + \log (| S_{n, p_{n}} |) \\ - 2 \log L_{m} (Y, θ θ) & = N_{0} \log 2 π + \sum_{t = 1}^{n} \sum_{i = 1}^{q * p_{t}} (\log F_{t, i} + \frac{ν_{t, i}^{2}}{F_{t, i}}) - b_{n, p_{n}}^{^{'}} S_{n, p_{n}}^{- 1} b_{n, p_{n}} + \log (| S_{n, p_{n}} |) \\ - \log (| S_{n, p_{n}}^{*} |) \\ - 2 \log L_{p} (Y, θ θ) & = N \log 2 π + \sum_{t = 1}^{n} \sum_{i = 1}^{q * p_{t}} (\log F_{t, i} + \frac{ν_{t, i}^{2}}{F_{t, i}}) - b_{n, p_{n}}^{^{'}} S_{n, p_{n}}^{- 1} b_{n, p_{n}} \end{aligned}

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\begin{equation*} \begin{aligned} -2 \log \mb{L}_{d}( \mb{Y}, \pmb {\theta } ) & = N_{0} \log 2 \pi + \sum _{t=1}^{n} \sum _{i=1}^{q*p_{t}} ( \log F_{t, i} + \frac{\nu _{t,i}^{2} }{ F_{t, i} } ) - \mb{b}_{n, p_{n}}^{'} \mb{S}_{n, p_{n}}^{-1} \mb{b}_{n, p_{n}} + \log ( | \mb{S}_{n, p_{n}} | ) \nonumber \\ -2 \log \mb{L}_{m}( \mb{Y}, \pmb {\theta } ) & = N_{0} \log 2 \pi + \sum _{t=1}^{n} \sum _{i=1}^{q*p_{t}} ( \log F_{t, i} + \frac{\nu _{t,i}^{2} }{ F_{t, i} } ) - \mb{b}_{n, p_{n}}^{'} \mb{S}_{n, p_{n}}^{-1} \mb{b}_{n, p_{n}} + \log ( | \mb{S}_{n, p_{n}} | ) \nonumber \\ & \; \; \; \; - \; \log ( | \mb{S}_{n, p_{n}}^{*} | ) \nonumber \\ -2 \log \mb{L}_{p}( \mb{Y}, \pmb {\theta } ) & = N \log 2 \pi + \sum _{t=1}^{n} \sum _{i=1}^{q*p_{t}} ( \log F_{t, i} + \frac{\nu _{t,i}^{2} }{ F_{t, i} } ) - \mb{b}_{n, p_{n}}^{'} \mb{S}_{n, p_{n}}^{-1} \mb{b}_{n, p_{n}} \nonumber \end{aligned}\end{equation*}