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\begin{aligned} E ({\hat{u}}_{f}^{^{'}} P {\hat{u}}_{f}) & = (M - N - T - K + 1) σ_{e}^{2} \\ E ({\tilde{u}}_{λ}^{^{'}} {\tilde{u}}_{λ}) & = (M - T - K) σ_{e}^{2} + [M - T - t r {X^{^{'}} W_{λ} D_{N} D_{N}^{^{'}} W_{λ} X {(X^{^{'}} W_{λ} X)}^{- 1}}] σ_{ν}^{2} \\ E ({\tilde{u}}_{ν}^{^{'}} {\tilde{u}}_{ν}) & = (M - N - K) σ_{e}^{2} + [M - N - t r {X^{^{'}} W_{ν} D_{T} D_{T}^{^{'}} W_{ν} X {(X^{^{'}} W_{ν} X)}^{- 1}}] σ_{λ}^{2} \end{aligned}

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\begin{align*} E\left(\hat{\mb{u}}_ f^{'} \mb{P} \hat{\mb{u}}_ f\right) & = (M - N - T - K + 1) \sigma ^2_ e \\ E\left(\tilde{\mb{u}}_\lambda ^{'} \tilde{\mb{u}}_\lambda \right) & = (M - T - K)\sigma ^2_ e + \left[ M - T - \mr{tr}\left\{ \mb{X}^{'} \mb{W}_\lambda \mb{D}_ N\mb{D}_ N^{'} \mb{W}_\lambda \mb{X} \left(\mb{X}^{'}\mb{W}_\lambda \mb{X}\right)^{-1}\right\} \right]\sigma ^2_\nu \\ E\left(\tilde{\mb{u}}_\nu ^{'} \tilde{\mb{u}}_\nu \right) & = (M - N - K)\sigma ^2_ e + \left[ M - N - \mr{tr}\left\{ \mb{X}^{'}\mb{W}_\nu \mb{D}_ T \mb{D}_ T^{'}\mb{W}_\nu \mb{X} \left(\mb{X}^{'}\mb{W}_\nu \mb{X}\right)^{-1}\right\} \right] \sigma ^2_\lambda \end{align*}