\begin{align}
\mathbb{B}\big(\hat{\boldsymbol{\beta}}^{PTE}\big)& =\mathbb{E}\big(\sqrt{n}(\hat{\boldsymbol{\beta}}^{PTE}-\boldsymbol{\beta})\big) \nonumber \\
& = \mathbb{E} \bigg[\sqrt{n} \bigg\{\hat{ (\boldsymbol{\beta}}^{UN}- \boldsymbol{\beta}) - (\hat{\boldsymbol{\beta}}^{UN}-\hat{\boldsymbol{\beta}}^{RE}) I_{(F_n \leq \chi^2_{(r, \alpha)})} \bigg\} \bigg] \nonumber\\
& =\mathbb{E} \bigg[Z_1- Z_3 I_{(F_n \leq \chi^2_{(r, \alpha)})} \bigg] \nonumber\\
& = \mathbb{E}  (Z_1)- \mathbb{E} \bigg[Z_3 I_{(F_n \leq \chi^2_{(r, \alpha)})} \bigg] \nonumber\\
& =- \mathbb{E}  (Z_3)\mathbb{E} \bigg[I_{(F_n \leq \chi^2_{(r, \alpha)})} \bigg] \nonumber\\
& = - \boldsymbol{\kappa} \boldsymbol{\gamma}\Upsilon_{r+2}(\chi^2_{(r, \alpha)}; \delta)\nonumber
\end{align}
